SOLUTION: Solve the problem. underbrace(5 + 8 + 11 + 14 + . . . + 299 + 302) = 100 terms

Question 1203626: Solve the problem.
underbrace(5 + 8 + 11 + 14 + . . . + 299 + 302) =
100 terms
Found 2 solutions by mananth, greenestamps:
Answer by mananth(16946) (Show Source): You can put this solution on YOUR website!
(5 + 8 + 11 + 14 + . . . + 299 + 302) =
This is an arithmetic sequence because the common difference between each consecutive term is constant The sum of an arithmetic sequence is given by the formula:
Sn= (n/2) * [2a + (n-1)d]
n is the number of terms.
a is the first term. -------------a= 5
d is the common difference ------- d=3
First we have to find the number of terms in the sequence ,n
To find n, we use the formula for the nth term of an arithmetic sequence:
nth term = a + (n-1)d
Plug values
302 = 5 + (n-1) * 3
302 - 5 = (n-1) * 3
297 = (n-1) * 3
Divide by 3
99 = n - 1
So n = 100
There are 100 terms in this serquence. n=100
Now, put these values into the sum formula:
Sn = (100/2) * [2 * 5 + (100-1) * 3]
Sn = (50) * [10 + 99 * 3]
Sn = 50 * (10 + 297)
Sn = 50 * 307
Sum = 15,350
Therefore, the sum of the given terms is 15,350.

Answer by greenestamps(13200) (Show Source): You can put this solution on YOUR website!

The formula found in many (most) references for the sum of the terms of an arithmetic sequence is this:

S(n) = (n/2)(2a+(n-1)d)

I dislike teaching this formula because it is like "magic" -- it is not at all clear why the formula works.

I much prefer teaching finding the sum of the terms of an arithmetic sequence using a formula that is easily understood:

S(n) = (number of terms) * (average of terms)

So for any particular problem like this, we need to be able to find the number of terms and the average of the terms.

Finding the average of the terms in an arithmetic sequence is easy. Because of the equal spacing of the terms, the average of all the terms is the average of the first and last terms.

So for this problem, the average of all the terms is (5+302)/2 = 307/2.

For a typical problem like this, finding the number of terms requires a bit more work. But in this problem the number of terms is given.

But let's suppose it was not given to us and we had to determine it. For this problem....

302-5 = 297 that's the difference between the first and last terms

297/3 = 99 since the common difference is 3, 99 is the number of terms in the sequence AFTER THE FIRST ONE.

So the number of terms in the sequence is 99+1 = 100.

Then the calculation for the sum of the terms in this problem is

S(n) = 100(307/2) = 100(153.5) = 15350

ANSWER: 15350

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