**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, S_{7} = (2 a + (7 – 1) d) = 7 (a + 3 d)

It is given that Sum of first 7 terms, S_{7} = 49

∴ 7 (a + 3 d) = 49

∴ (a + 3 d) = 7 ……. (i)

**Step 2:**

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

It is given that Sum of first 7 terms, S_{7} = 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, S_{20} = (2 (1) + (20 – 1) (2))

= 10 (2 + 38)

= 400

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

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