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,
letβs recall some key facts about any geometric sequence. The first term of any geometric
sequence is denoted by the letter π. The common ratio of any geometric
sequence is equal to π. This means that the terms of a
geometric sequence are π, ππ, ππ squared, and so on. The sum to β of any geometric
sequence is equal to π divided by one minus π. This is only true though if the
modulus or absolute value of π is less than one.
In the geometric sequence in this
question, π is equal to 87. ππ is equal to the square root of
87. We can calculate the value of π,
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 and π into the formula for the sum to β.
The sum to β 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.
