Q) If the sum of first 7 terms of an A.P. is 49 and that of first 17 terms is 289, find the sum of its first 20 terms.

Ans: 

Let’s consider the first term be A  and the common difference be D.

Step 1: We know that sum of first n terms of an A.P.  S_n = (2 a + (n – 1) d)

Therefore, Sum of first 7 terms, S₇ = (2 a + (7 – 1) d) = 7 (a + 3 d)

It is given that Sum of first 7 terms, S₇ =  49

∴ 7 (a + 3 d) = 49

∴ (a + 3 d) = 7  ……. (i)

Step 2:

Sum of first 17 terms, S₁₇ = (2 a + (17 – 1) d) = 17 (a + 8 d)

It is given that Sum of first 7 terms, S₇ =  289

∴ 17 (a + 8 d) = 289

∴ (a + 8 d) = 17  ……. (ii)

Step 3: By subtracting equation (i) from (ii), we get:

(a + 8 d) – (a + 3 d) = 17 – 7

∴ 8 d – 3 d = 10

∴ d =

∴ d = 2

from equation (i), we have a + 3 d = 7

by substituting value of d = 2 in this equation, we get:

a + 3 (2) = 7

∴ a = 7 – 6

∴ a = 1

Step 4: Sum of first n terms of an A.P.  S_n = (2 a + (n – 1) d)

Now we have a = 1, d = 2, n = 20

∴ Sum of first 20 terms, S₂₀ = (2 (1) + (20 – 1) (2))

= 10 (2 + 38)

= 400

Therefore, the Sum of first 20 terms of AP is 400.

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