Derivation of Geometric Sum Formula
The sum of a geometric series up to a finite number of terms is derived as follows. Let us call Sn as the sum of a geometric series up to n terms. Then, we have,
Sn = a + ar + ar2 + ar3 + . . . + arn-1
⇒ Sn = a × (1 + r + r2 + r3 + . . . + rn-1)
We have a relation that,
1 – rn = (1 – r) × (1 + r + r2 + r3 + . . . + rn-1)
Using this relation, we get,
Sn = a × (1 – rn)/(1 – r)
Where,
- a is the first term
- r is the common ratio
- n is the number of terms
How to Find the Sum of Geometric Series
A geometric series is a sequence of numbers where each term after the first term is found by multiplying the previous term by a fixed, non-zero number called the common ratio. In a geometric series, if the absolute value of the common ratio (∣r∣) is less than 1, the series converges to a finite value. Otherwise, it diverges (grows without bound). Let’s know more about sum of Geometric Series formula, derivation and examples in detail below.
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