Question 550225: a,b,b,c,c,c,d,d,d,d,e,e,e,e,e.....................find the 280th term of the series...Answer choices w,y,x,u Answer by bucky(2189) (Show Source):
You can put this solution on YOUR website! I'm not sure that this is the best approach, but here's an algebraic way to solve this problem.
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First, note that each change in the series follows an arithmetic pattern. The first term is just the single letter "a". The next group of terms is two of the letter b. The next group of terms are three of the letter c. The next group of terms are four of the letter d. And so on. This means that the total number of letters at any point in the series can be found by adding:
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1 + 2 + 3 + 4 + ........
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Here's a simple check that might clarify this a bit. How many total letters are in the series at the end of the fifth letter which is e? The answer to that is:
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1 + 2 + 3 + 4 + 5 = 15
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And if you count all the letters in the opening of the sample series given in the problem, you find there are a total of 15 letters ... one a plus 2 b + 3 c + 4 d + 5 e = 15.
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In other words, each number in this example series represents a letter. There are 5 numbers (1 through 5) so each number corresponds to a letter (a through e). The value of the number tells how many of each corresponding letter appears in the series. You need to understand this. Maybe the further explanation below will help.
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Notice that this is an arithmetic series. You may recall that the sum (call it S) of the terms in an arithmetic series where each term increases by 1 from the previous term is given by the equation:
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in which n represents the number of terms in the series. And for this problem n REPRESENTS THE NUMBER OF SUCCESSIVE GROUPS OF ALPHABET LETTERS THAT HAVE BEEN USED.
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You need to find out what number n will give you the 280th term, and that can be done by setting the S (for sum) equal to 280 in the equation as follows:
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Multiply both sides by the denominator 2 on the right side and you have:
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Multiply out the right side to get:
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Subtract 560 from both sides:
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And transpose to get the conventional quadratic form of:
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Note that for this problem, you are only interested in positive values for n.
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Apply the quadratic formula to solve for n. The value for "a" in the formula (the multiplier of the n-squared term) is +1. The value for "b" in the formula (the multiplier of the n-term) is also +1. And the value for "c" (the constant) is = -560. Substitute those values into the quadratic formula:
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I'll leave the math work to you. Remember you are only interested in a positive answer for n. You can disregard the negative n.
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After you do the work you should get an answer of n = 23.17
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Think about what this means. You have to go out more than 23 letters to get to the 280th term of the series you were given by the problem. The 23rd letter of the alphabet is w. But you need to go beyond w and into the group of x's. (The end of the x's occurs when n = 24.00). So the answer to this problem is the 280th term is in the group of x's, so the letter x is the 280th term.
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Sort of difficult to explain, but I hope this gives you a way to view how this problem can be solved. And I hope this helps.
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