### Solution Found!

# Derive the formula (1.4) for the sum Sn of the geometric progression Sn = a + ar + ar2 +

**Chapter 1, Problem 2**

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**QUESTION:**

Derive the formula (1.4) for the sum Sn of the geometric progression Sn = a + ar + ar2 + + arn1. Hint: Multiply Sn by r and subtract the result from Sn; then solve for Sn. Show that the geometric series (1.6) converges if and only if |r| < 1; also show that if |r| < 1, the sum is given by equation (1.8).

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### Questions & Answers

**QUESTION:**

Derive the formula (1.4) for the sum Sn of the geometric progression Sn = a + ar + ar2 + + arn1. Hint: Multiply Sn by r and subtract the result from Sn; then solve for Sn. Show that the geometric series (1.6) converges if and only if |r| < 1; also show that if |r| < 1, the sum is given by equation (1.8).

**ANSWER:**

Step 1 of 2

Given

(i)

To derive a formula for .

Multiplying with the both sides of equation (i),

(ii)

Subtracting equation (ii) from equation (i),

Thus, the sum of a geometric series with first term and common ratio is formulated by