Which formula can be used to sum the first n terms of a geometric sequence?

Solution:

A geometric progression is a sequence where every term bears a constant ratio to its preceding term.

The formula used to calculate the sum of 'n' terms of a geometric sequence is given as,

S_n = a(1 - rⁿ) / (1 - r), where r is not equal to 1.

Here,

• 'a' is the first term
• 'r' is the common ratio
• 'n' is the number of terms
• 'S_n' is the sum of 'n' terms

Let's take an example to understand this.

Here is a GP sequence 3, 15, 75, …. Let us calculate the sum of the first 7 terms.

a = 3, r = 5, n = 7

⇒ S_n = a (1 - rⁿ) / (1 - r)

⇒ S₇ = 3 (1 - 5⁷ ) / (1 - 5)

⇒ S₇ = 3 × (1 - 78125) / ( - 4)

⇒ S₇ = 3 × ( - 78124) / ( - 4)

⇒ S₇ = 3 × 19531

⇒ S₇ = 58593

Thus, the sum of the first 7 terms of the given GP is 58593.

Hence, the sum of 'n' terms of a GP is calculated using S_n = a(1 - rⁿ) / (1 - r)

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Which formula can be used to sum the first n terms of a geometric sequence?

Summary:

The formula that is used to find the sum the first n terms of a geometric sequence is S_n = a(1 - rⁿ) / (1 - r)