A LEVEL MATHEMATICS

Question

An infinite geometric series has common ratio r. the sum to infinity of the series is 7 times the first term of the series. Given that the first term of the series is 15, find the least value of n such that the nth term of the series is less than 1.

Answer 1

Hi Tea - Good Question!

The sum of an infinite geometric series is as follows:

Sum = A₁/(1 - r)

Where A₁ equals the first term in the series and r equals the common ratio.

The problem states that the sum is equal to 7 times the first term (so 7A₁). We use this information to solve for r:

7A₁ = 15/(1 - r)

7(15) = 15/(1 - r) Divide both sides by 15

7 = 1/(1 - r)

1/7 = 1 - r

r = 6/7

Now that we have solved for r, we can find out how many terms it takes for the nth term to be less than 1 by solving 15 x (6/7)^n-1 < 1 for n.

After using a calculator, we see that n = 19