Calculating the Sum of an Infinite Geometric Sequence | Formula and Conditions

The Geometric Sequence Infinite Sum

To find the sum of an infinite geometric sequence, we can use the formula:

S = a / (1 – r)

Where:
– S represents the sum of the sequence
– a is the first term of the sequence
– r is the common ratio between consecutive terms

However, it’s important to note that this formula only holds true if the common ratio (r) is between -1 and 1

To find the sum of an infinite geometric sequence, we can use the formula:

S = a / (1 – r)

Where:
– S represents the sum of the sequence
– a is the first term of the sequence
– r is the common ratio between consecutive terms

However, it’s important to note that this formula only holds true if the common ratio (r) is between -1 and 1. If the common ratio is outside this range, the sequence doesn’t converge to a finite sum and we can’t apply this formula.

Let’s understand why this formula works by considering an infinite geometric sequence:

a, ar, ar^2, ar^3, …

To find the sum of this sequence, let’s multiply it by the common ratio (r):

r(a, ar, ar^2, ar^3, …) = ar, ar^2, ar^3, ar^4, …

Now, if we subtract the second sequence from the first, we can cancel out a common term:

(a, ar, ar^2, ar^3, …) – (ar, ar^2, ar^3, ar^4, …) = a

By subtracting the two sequences, we eliminated all the terms except for the first term (a). Now, let’s divide this equation by (1 – r):

(a / (1 – r)) = a / (1 – r)

On the left side, we have the sum of the original sequence (S), and on the right side, we have the remaining term (a / (1 – r)). Hence, we can conclude that:

S = a / (1 – r)

Only when the common ratio (r) is between -1 and 1 can the above formula be used to find the sum of an infinite geometric sequence. Otherwise, the sequence does not converge to a finite sum.

In summary, to calculate the sum of an infinite geometric sequence, use the formula S = a / (1 – r), but ensure that the common ratio (r) is within the range -1 to 1.

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