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If the sum of the first 7 terms of an arithmetic progression (AP) is 119, and the sum of the first 17 terms is 714, find the sum of the first n terms of the AP.

A) Use the formula for the sum of n terms in an AP.
B) Determine the common difference of the AP.
C) Find the value of 'n' using the given information.
D) Check for arithmetic progression properties.

Answer :

Final answer:

To find the sum of the first n terms of an arithmetic progression (AP), we can use the formula: S_n = (n/2)(a_1 + a_n), where S_n is the sum of the n terms, a_1 is the first term, and a_n is the nth term. Given the values of the sum of the first 7 terms and the sum of the first 17 terms, we can calculate the common difference (d) and the first term (a_1).

Explanation:

To find the sum of the first n terms of an arithmetic progression (AP), we can use the formula:

Sn = (n/2)(a1 + an)

where Sn is the sum of the n terms, a1 is the first term, and an is the nth term. To find the sum of the first n terms of the given AP, we need to determine the values of a1 and an.

Given that the sum of the first 7 terms of the AP is 119, we can use the formula to calculate the common difference (d):

119 = (7/2)(a1 + a7)

Similarly, given that the sum of the first 17 terms of the AP is 714, we can calculate the value of a1 + 16d (which is equal to a17):

714 = (17/2)(a1 + a17)

By solving these two equations, we can find the values of a1, a7, and d.

Once we have the values of a1 and d, we can use the formula to find the sum of the first n terms of the AP for any value of n.

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