Question

Find the sum of the finite geometric sequence where the first term=4,last term=1024,r=-2

Answer 1

Answer:

We will learn how to find the sum of n terms of the Geometric Progression {a, ar, ar2, ar3, ar4, ...........}

To prove that the sum of first n terms of the Geometric Progression whose first term ‘a’ and common ratio ‘r’ is given by

Sn = a(rn−1r−1)

⇒ Sn = a(1−rn1−r), r ≠ 1.

Let Sn denote the sum of n terms of the Geometric Progression {a, ar, ar2, ar3, ar4, ...........} with first term ‘a’ and common ratio r. Then,

Now, the nth terms of the given Geometric Progression = a ∙ rn−1.

Therefore, Sn = a + ar + ar2 + ar3 + ar4 + ............... + arn−2 + arn−1 ............ (i)

Multiplying both sides by r, we get,

rSn = ar + ar2 + ar3 + ar4 + ar4 + ................ + arn−1 + arn ............ (ii)

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On subtracting (ii) from (i), we get

Sn - rSn = a - arn

⇒ Sn(1 - r) = a(1 - rn)

⇒ Sn = a(1−rn)(1−r)

⇒ Sn = a(rn−1)(r−1)

Hence, Sn = a(1−rn)(1−r) or, Sn = a(rn−1)(r−1)