Video: Finding the Sum of an Infinite Number of Terms in a Given Geometric Sequence

Find the sum of an infinite number of terms of the geometric sequence (87, √(87), 1, ...).

03:23

Video Transcript

Find the sum of an infinite number of terms of the geometric sequence. 87, square root of 87, one, and so on.

Before starting this question, let’s recall some key facts about any geometric sequence. The first term of any geometric sequence is denoted by the letter π‘Ž. The common ratio of any geometric sequence is equal to π‘Ÿ. This means that the terms of a geometric sequence are π‘Ž, π‘Žπ‘Ÿ, π‘Žπ‘Ÿ squared, and so on. The sum to ∞ of any geometric sequence is equal to π‘Ž divided by one minus π‘Ÿ. This is only true though if the modulus or absolute value of π‘Ÿ is less than one.

In the geometric sequence in this question, π‘Ž is equal to 87. π‘Žπ‘Ÿ is equal to the square root of 87. We can calculate the value of π‘Ÿ, the common ratio, by dividing these two terms. The common ratio is therefore the square root of 87 over 87. We can now substitute our values for π‘Ž and π‘Ÿ into the formula for the sum to ∞.

The sum to ∞ is equal to 87 divided by one minus the square root of 87 over 87. In order to simplify this fraction, we need to rationalise the denominator. We multiply the numerator and denominator by one plus the square root of 87 over 87. 87 multiplied by one is equal to 87. 87 multiplied by the square root of 87 over 87 is equal to the square root of 87.

On the denominator, we have the difference of two squares. This means that we firstly need to multiply the first terms, one multiplied by one. This is equal to one. We also need to multiply the last terms, negative square root of 87 over 87 and positive square root of 87 over 87. This simplifies to negative one over 87. One whole one is the same as 87 over 87. As the denominators here are the same, we can subtract the numerators. This gives us 86 over 87. Dividing by a fraction is the same as multiplying by the reciprocal of this fraction. The reciprocal of 86 over 87 is 87 over 86.

We can therefore conclude that the sum of an infinite number of terms of the geometric sequence is 87 over 86 multiplied by 87 plus the square root of 87.

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