Question 1101665: the 15th term in an arithmetic sequence is 129 and the sum of the first fifteen terms is 1095. find the sum if the first ten terms. Found 2 solutions by josgarithmetic, greenestamps:Answer by josgarithmetic(39618) (Show Source):
You can put this solution on YOUR website! A, the first term
The 15th term is .
Sum of first 15 terms is .
Two linear equations in A and d.
First, a few general facts about problems like this....
(1) The average of any set of numbers is the sum of all the numbers, divided by how many numbers there are.
(2) So the sum of any set of numbers can be thought of as the product of the average of the numbers and how many there are.
(3) In an arithmetic sequence, because the numbers are equally spaced, the average of all the numbers is the average of the first and last numbers.
(4) So in an arithmetic sequence, the sum of the first n terms can be calculated as (number of terms) times (average of first and last terms).
In this problem, since we are given both the last (15th) term and the sum of the 15 terms, we can use (4) above to immediately find the first number in the sequence. Then it is easy work to find the common difference and then the sum of the first 10 terms.
sum = (how many there are) times (average of first and last):
The first term is 17. Use that and the given value of the 15th term to find the common difference.
The 15th term is the first term, plus the common difference 14 times:
The common difference is 8. Use that and the first term to find the 10th term in the sequence.
The 10th term is the first term, plus the common difference 9 times:
The 10th term is 89.
The sum of the first 10 terms is the number of terms times the average of the first and tenth terms:
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