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# Recent problems solved by 'nabla'

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 Sequences-and-series/287702: write the series with summation notation. 5/6+6/7+7/8+8/9+....1 solutions Answer 208560 by nabla(475)   on 2010-04-01 23:05:17 (Show Source): You can put this solution on YOUR website!Sum from n=0 to infinity (5+n)/(6+n)
 Sequences-and-series/287489: For questions 1–6 identify each sequence as arithmetic or geometric. Then find the common difference or common ratio. 1. 13, 19, 25, 31, ... 2. 16, 24, 36, 54, ... 3. 4, 14, 24, 34, ... 4. 1, 4, 16, 64, ... 5. 35, 33, 31, 29, ... 6. 64, 48, 36, 27, ... Thanks!1 solutions Answer 208330 by nabla(475)   on 2010-03-31 22:40:11 (Show Source): You can put this solution on YOUR website!Arithmetic means each term changes based on adding or subtracting. Geometric means each term changes based on multiplication (or division). 1. You are adding 6 to get to each successive term. 2. You are multiplying by 3/2 to get each successive term. 3. You are adding 10 to get to each successive term. 4. You are multiplying by four to get each successive term. 5. You are subtracting 2 to get each successive term. 6. You are multiplying by 3/4 to get each successive term. Apply the rule I established above and this problem is finished!
 Trigonometry-basics/287526: Find a function f such that the graph of f has a horizontal tangent at (2, 0) and f ''(x) = 4x. f(x) = ?1 solutions Answer 208328 by nabla(475)   on 2010-03-31 22:28:06 (Show Source): You can put this solution on YOUR website!Integrate (find the antiderivative) twice: f'(x)=2x^2+C f(x)=(2/3)x^3+Cx+D Now, f has a horizontal tangent at (2,0) IE the function attains a local extrema and f'(2)=0 hence f'(2)=2*2^2+C=0 implies C=-8. Now we have f(x)=(2/3)x^3-8x+D. But we know f(2)=0, so, (2/3)2^3-8(2)+D=0 16/3-48/3=-D -32/3=-D implies 32/3=D. This gives our final function as f(x)=(2/3)x^3-8x+32/3.
 Surface-area/287497: what is the surface area of a pyramid?1 solutions Answer 208326 by nabla(475)   on 2010-03-31 22:21:20 (Show Source): You can put this solution on YOUR website!For a square pyramid (ie the base is a square...): Surface Area = [Base Area] + 1/2 × Perimeter × [Side Length] See http://www.mathsisfun.com/geometry/square-pyramid.html
 Signed-numbers/287433: Determine the following integral: The aria bound by the straight line y=x and tha parabola 1 solutions Answer 208308 by nabla(475)   on 2010-03-31 21:07:56 (Show Source): You can put this solution on YOUR website!First, find where x=x^2 (this gives the end points of the closed interior). This occurs when 1=x and 0=x. So, we are interested in the closed interval [0,1]. Now, you will note by graphing that x>=x^2 throughout [0,1]. IE. .5>.25 etc. So we integrate from 0 to 1 of (x-x^2)dx. This is (x^2/2) -(x^3/3) evaluated from 0 to 1 so we have (1/2)-(1/3)=1/6.
 Numbers_Word_Problems/286600: The number 9 can be written as the sum of nine consecutive integers. What is the product of these intergers? 1 solutions Answer 207897 by nabla(475)   on 2010-03-29 20:17:33 (Show Source): You can put this solution on YOUR website!9 consecutive integers: n,n+1,n+2,n+3,n+4,n+5,n+6,n+7,n+8 So their sum is 9n+36=9(n+4) So, we want 9(n+4)=9 this implies n+4=1 and n=-3. So, -3+-2+-1+0+1+2+3+4+5=9. Now, one should note that 0 is included, so the product of any numbers with 0 has to be 0.
 Divisibility_and_Prime_Numbers/286011: Is the following set of numbers a perfect triple? 6, 8, 101 solutions Answer 207469 by nabla(475)   on 2010-03-28 14:11:52 (Show Source): You can put this solution on YOUR website!If by perfect triple is meant Pythagorean triple, we must have the following criteria met (there are more but these should be more than sufficient...): a^2+b^2=c^2 a=s^2-t^2 b=2st c=s^2+t^2 with s,t integers. (Note: a and b are interchangeable, IE it doesn't matter which comes first in the first equation.) There are also interesting properties of s and t that one could investigate if interested. Additionally, |a+b|<=|a|+|b|. IE the hypotenuse is less than or equal to the sum of the legs. This criteria is met by your three numbers, so we must investigate further with the first equation [easiest to do]. The question is: is 6^2+8^2=10^2? IE is 36+64=100? Yes. The last two lines are sufficient to understand the concept, but I hope you see there are many more relationships to note with Pythagorean triples than may be presented in a basic algebra or geometry text.
 Graphs/286111: find 3 numbers that their mode , median and mean are equal1 solutions Answer 207468 by nabla(475)   on 2010-03-28 14:01:20 (Show Source): You can put this solution on YOUR website!Any 3 of the same number will work. Also, we can set up something like this S={2,4,6,} Then S is multimodal, its median is 4, and its mean is 12/3=4. So we can argue that this set will work, but the multimodality gives something "extra" so it is strictly not perfectly equal.
 real-numbers/286032: prove that no number of the type 4k+2 be a perfect square.1 solutions Answer 207465 by nabla(475)   on 2010-03-28 13:53:02 (Show Source): You can put this solution on YOUR website!Denote a^2=4k+2=2(2k+1). So 2|a^2 thus 2|a. Let a=2n. Then a^2=4n^2 which means 4|a^2. However, we have a^2=4k+2. 4 does not divide 4k+2, hence 4k+2 can never be a perfect square. In other words: if a is even, its square will always be divisible by 4.
 Exponents/285781: Solve for x: (x + 1)^3/2 = 8 A. 3 B. 5 C. 21.63 D. 25 E. None of these.1 solutions Answer 207215 by nabla(475)   on 2010-03-27 08:41:04 (Show Source): You can put this solution on YOUR website!Take each side of the equation to the 2/3. 8^(2/3)=4. So we want x+1=4->x=3. Answer A.
 logarithm/285745: Choose the expression or equation that is equivalent to the one given. log3 5 - log3 91 solutions Answer 207214 by nabla(475)   on 2010-03-27 08:39:41 (Show Source): You can put this solution on YOUR website!This is equivalent to log3 (5/9) in your notation. When you subtract logarithms with same base you divide the numbers inside a log.
 absolute-value/285792: The number of real solutions of the equation | x – 2 | + | x – 3 | = 3 is A. 1 B 2 C 3 D 4 E None of these1 solutions Answer 207213 by nabla(475)   on 2010-03-27 08:36:33 (Show Source): You can put this solution on YOUR website!You are asking when 2x-5=3 or 2x-5=-3. IE 2x=8->x=4 or 2x=2->x=1. Two solutions, B.
 Permutations/285761: how many positive integers less than or equal to 70,are relatively prime 70?1 solutions Answer 207212 by nabla(475)   on 2010-03-27 08:31:01 (Show Source): You can put this solution on YOUR website!Use Euler's phi function. IE phi(n)=n(1-p_1)(1-p_2)...(1-p_r) where each p_i is a constituent prime (powers do not matter). 70=35*2=7*5*2 phi(70)=70(6/7)(4/5)(1/2)=6*4=24.
 Problems-with-consecutive-odd-even-integers/285791: Consider the integers from 1 to 100 inclusive. What is the difference between the sum of all the even integers and the sum of all the odd integers? A. 1 B. 2 C. 0 D. 4 • E. 501 solutions Answer 207209 by nabla(475)   on 2010-03-27 08:20:09 (Show Source): You can put this solution on YOUR website!Let's denote our odds as: SUM(k=0->49)[2k+1] and our evens as: SUM(k=0->50)[2k] So, we want to evaluate: SUM(k=0->50)[2k]-SUM(k=0->49)[2k+1] this is equivalent to: 2*SUM(k=0->50)[k]-2SUM(k=0->49)[k]-50 which is moreover equivalent to: 2*50+2*SUM(k=0->49)[k]-2SUM(k=0->49)[k]-50 Our summation notation cancels, and we are left with 100-50=50. So we shall pick E as the solution. If these properties of series are unfamiliar, note first that SUM(k=0->49)[1] is 1 added up 50 times (=50). Moreover, When we easily remove first or last members of the series by simply adding the term, hence 2*SUM(k=0->50)[k]=2*50+SUM(k=0->49)[k]. Hope this helps.
 Quadratic_Equations/285634: Having trouble solving , tried the problem solver and got a particularly complicated solution that I can't prove.1 solutions Answer 207049 by nabla(475)   on 2010-03-26 15:49:45 (Show Source): You can put this solution on YOUR website! For polynomial ax^2+bx+c=0. You have a=3, b=5, c=-6. You plug those into the equation above and simplify radicals as necessary. (1/6)(-5+/-sqrt(97)
 Numbers_Word_Problems/285523: How many whole numbers lie in the interval between 5/3 and 2π ? (A) 2 (B) 3 (C) 4 (D) 5 (E) infinitely many1 solutions Answer 207048 by nabla(475)   on 2010-03-26 15:45:49 (Show Source): You can put this solution on YOUR website!5/3=1.66..., 2pi is "about 2*3.14"=6.28. So, we've got 2,3,4,5,6. 5 numbers D.
 Polynomials-and-rational-expressions/283339: Is the product of a binomial always a trinomial? Explain with examples.1 solutions Answer 205734 by nabla(475)   on 2010-03-21 12:22:35 (Show Source): You can put this solution on YOUR website!Is the question this: Is the product of two binomials always a trinomial? If not, disregard this solution. Consider: (a+b)(a-b)=a^2-ab+ab-b^2=a^2-b^2. This is a binomial. Hence, it is not always true that the product of two binomials is a trinomial. However, it is true in many other cases: (a+b)^2=a^2+2ab+b^2
 Polynomials-and-rational-expressions/283359: If x^2 = 3x + 4, then x^5 is equal to: (A) 3x + 4 (B) 13x + 12 (C) 51x + 52 (D) 205x + 204 (E) 819x + 8201 solutions Answer 205731 by nabla(475)   on 2010-03-21 12:14:07 (Show Source): You can put this solution on YOUR website!You solve this by substituting x^2 into the expansion of x^5. ---> D is indeed the answer, but I hope you now see how to do it.
 Numbers_Word_Problems/283356: Which of the following numbers can be expressed as a sum of 2 or more consecutive odd whole numbers? i) 225 ii) 400 a) (i) only b) (ii) only c) Both (i) and (ii) d) none of these1 solutions Answer 205719 by nabla(475)   on 2010-03-21 11:04:14 (Show Source): You can put this solution on YOUR website!Consider: Odd numbers can be expressed as: 2k+1 (k a natural number) Hence consecutive odds will be 2k+1 and 2k+3. The sum of these two is 4k+4. Hence, the number must be even so case i is out. 4k+4=400 implies k=99. So, 400 works. However, as per the tutor below there are additional cases to consider. Now consider 3 odds. 2k+1,2k+3,2k+5. This gives 6k+9 as the sum. 6k+9=225 -> 6=16. So 225 fits the "or more" portion of the question. I hope this helps you see how to work the problem rather than just supplying the answer...
 Linear-equations/283028: Determine whether 2 is the solution of the equality: x>-2 yes or no1 solutions Answer 205530 by nabla(475)   on 2010-03-20 10:52:50 (Show Source): You can put this solution on YOUR website!The question is this: Is 2>-2? In words: is 2 bigger than -2? We can add numbers to both sides and retain the inequality. So the question is the same as: Is 2+2>-2+2? Rather, is 4>0? If you have four of something, don't you have more than if you had none of something? You can also draw a number line. -2 -1 0 1 2 The number line by definition adheres to a property called induction which retains the situation that n < n+1
 Quadratic_Equations/283005: Solve by factoring and using the principle of zero products. 3x^2 + 24x = 01 solutions Answer 205528 by nabla(475)   on 2010-03-20 10:47:58 (Show Source): You can put this solution on YOUR website!Take out the greatest factors of each term: in this case it is 3x. 3x(x+8)=0 This means 3x=0 or x+8=0 which gives x=0 or x=-8 as solutions.
 Graphs/283096: How do I graph on a plane: -2x + 4y < 12 I have absolutely no idea what to do!! Thank you. Cynthia1 solutions Answer 205525 by nabla(475)   on 2010-03-20 10:44:06 (Show Source): You can put this solution on YOUR website!Rewrite for y (is easiest): y < x/2 +3 So, draw the function f(x)=x/2 +3 Now, we want y to always be less than that line (the line itself will be dotted!). So, we shade below it as pictured (fairly poorly but it is better than nothing):
 logarithm/283089: I feel stupid asking this. I'm just starting to work on these? The problem I got is log2y=3 Now if I'm correct, I try to find x and convert this to y=log(2)3 In which case, is it no solution? Or am I completely off in how I'm working on these? Thanks, Nathalie1 solutions Answer 205511 by nabla(475)   on 2010-03-20 08:31:04 (Show Source): You can put this solution on YOUR website!Is that a log of base 2? By definition log_2(y)=3 means 2^3=y implies 8=y. In words, the base of a log to the power of the other side of the equation is equal to what the log was of (this may be confusing and it certainly is not concise).
 Miscellaneous_Word_Problems/283083: How many real numbers are there such that the 5th power of the number is the sum of the 4th and 3rd powers of the number? a) 1 b) 2 c) 3 d) 5 e) none of these 1 solutions Answer 205503 by nabla(475)   on 2010-03-20 07:38:05 (Show Source): You can put this solution on YOUR website!So, we want: In other words we are asking when ? In even more words we are asking when ? We note that x=0 certainly makes the statement true. Now, will give two real (irrational) roots. So, strictly speaking there are three real numbers that meet the criterion: {0, 1/2 (1 - Sqrt[5]), 1/2 (1 + Sqrt[5]). However, one should note that x=0 occurs as a root three times.
 Probability-and-statistics/283076: What is the probability that the ones digit is 9 in the sum of two two-digit prime numbers? a) 4/10 b) 3/10 c) 0 d) 4/9 e) 1/3 1 solutions Answer 205501 by nabla(475)   on 2010-03-20 07:29:27 (Show Source): You can put this solution on YOUR website!There are 21 two digit prime numbers: 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 Can you make a 9 by adding any two of 1,3,7,9? Hence, one should choose c as the answer.
 Square-cubic-other-roots/283063: If what would the values of n and a be?1 solutions Answer 205499 by nabla(475)   on 2010-03-20 07:19:56 (Show Source): You can put this solution on YOUR website!First, simplify the left. So, we want 2=4n-6. This implies n=2. If n=2, a=3^(-1)=1/3. IE
 Numbers_Word_Problems/283081: The last digit of 7^1992 is a 1 b 2 c 6 d 7 e 9 1 solutions Answer 205494 by nabla(475)   on 2010-03-20 07:04:52 (Show Source): You can put this solution on YOUR website!You can figure this out a couple of ways, but I will show you how I would do it using modular arithmetic and congruences. First, 7^2=49 which is congruent to (-1) modulo 10. Second, 7^1992=(7^2)^996 which is congruent to (-1)^996 modulo 10. Finally, (-1)^996=1. We conclude that the unit digit of 7^1992 is 1. Without going through an entire explanation and lesson in modular congruences, I will explain this as follows: Consider: abcdefx where a,b,c,d,e,f,x are natural numbers that form the number abcdefx (we are not talking about multiplication here). Note that this is the same as abcdef0+x. Now, 10 divides a multiple of 10. So 10|(abcdef0). We are left with x mod 10 (as 10|abcdef0 implies abcdef0 is congruent to 0 mod 10). So when we take mod 10 of any number we can think of that as asking what the last digit will be. Hope this helps.
 Polynomials-and-rational-expressions/282094: find all of the rational zeros of this function f(x) = x^4 - 49x^21 solutions Answer 205090 by nabla(475)   on 2010-03-18 14:31:41 (Show Source): You can put this solution on YOUR website!Rewrite as: Use difference of squares formula. So we want: x^2+7x=0 or x^2-7x=0 ie: x(x+7)=0 or x(x-7)=0 x=-7 or 0; or x=7 or 0 Hence we have solution set {-7,0,7} with 0 occurring twice.
 Trigonometry-basics/282403: how do you figure out f(3.14) = tan (x/4) + sin (x) ? I have the answer but I don't understand how to figure it out. I keep getting it wrong.1 solutions Answer 205086 by nabla(475)   on 2010-03-18 14:18:49 (Show Source): You can put this solution on YOUR website!Please disregard this solution, I misread the question.
 Expressions-with-variables/282457: Is the ordered pair (-3, 6) a solution of 4x - 2y < 2? Could you please help me to solve this problem, step by step? Thank you. Cynthia1 solutions Answer 205085 by nabla(475)   on 2010-03-18 14:13:01 (Show Source): You can put this solution on YOUR website!It is asking you to put x=-3 and y=6. Put them into the inequality and see if it is true. 4(-3)-2(6)=-12-12=-24 Is -24 less than 2? Certainly. Hence, the ordered pair is a solution to the inequality.
 Finance/282470: What is the domain? p(x)=x^3-x^2-6. Is it the set {x|x is a real number and x does not equal 6}?1 solutions Answer 205082 by nabla(475)   on 2010-03-18 14:01:59 (Show Source): You can put this solution on YOUR website!Polynomials have all real numbers as their domains. The question of "what is the domain" is really one asking where is the function defined. Put 6 into the function. Do you get a "normal" function value? Yes, p(6)=174. For another example, consider f(x)=1/x. Here we know that the function behaves "funny" around x=0. Moreover, many people are aware of the fact that "we can not divide by zero." Although it takes a bit more math to see why this is true concisely, take a look at the graph and you will see some strange behavior. A final example, consider g(x)=ln(x). This is the natural log function. ln(x)=y means the same thing as e^y=x. Hence, x is always positive and not 0 as e^y=0 is nonsense.