#### The first three terms of an arithmetic series are p, 5p – 8, and 3p + 8 respectively.

Prove that the sum of the first n terms of the series is a perfect square.

Anonymous

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Prove that the sum of the first n terms of the series is a perfect square.

Krishna

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Step 1: Calculate the (any variable)p value

[NOTE:The definition of an arithmetic series is that the difference of two consecutive terms is always constant,]

EXAMPLE: p, 5p - 8, 3p+8

(5p - 8) - (p) = (3p + 8) - (5p - 8)

Step 2: Substitute the (any variable) p value in the given terms and simplify, to get the proper arithmetic series.

[RECALL: Simplify [Q-A1.O.6]

Step 4: Find the sum of the series.

Skill i: Identify the first term in the sequence, call this number a.

Skill ii: Calculate the common difference(d) of the sequence.

Skill iii: Identify the number of terms (n).

EXAMPLE: In A.P 2, 4, 6,..... find the sum of the first 10 terms. So take n=10

Skill iv: Plug the values of n, d, and a into the formula.

[FORMULA: The formula for finding the sum of first n terms of an

arithmetic sequence (S_n) =(n/2)[2a + (n- 1)d]

Skill v: Simplify further.(Apply the BODMAS rules)

Skill vi: Verify the result, is it satisfying the given statement

or not