Topic 1.8: Infinite Geometric Series Questions

Write down the sum to infinity.

Find the first term.

The infinite geometric series u1 + u2 + u3 + … converges to 8. Its common ratio is r.

Find u1 in terms of r.

The numbers $2,-6,18,-54,\dots$ form a geometric sequence.

State the reason that adding all of its terms cannot give a finite total.

A geometric series has first term $5$ and second term $7.5$.

State why the series has no sum to infinity.

A geometric sequence a1, a2, a3, … has common ratio r, where |r| < 1. The sum to infinity of the sequence is 12.

Find an expression for a1 in terms of r.

A geometric series has first term $\ln 9$ and second term $\ln 3$.

Show that the series has a sum to infinity.

Show that this series has a sum to infinity.

Show that the geometric series $5-\frac{5}{\sqrt{2}}+\frac{5}{2}-\dots$ converges.

Given that the sum to infinity is $27$, show that the common ratio is $\frac{1}{3}$, and state why the series converges.

A geometric series has first term 8 and a sum to infinity of 20.

The first term of a geometric series is 18 and the second term is 12.

A ball is dropped from 10 m and rebounds to ½ of its height each time. Find the total distance it travels before coming to rest.

Find the common ratio.

An infinite geometric series has first term $3$ and common ratio $\frac{x-1}{2}$.

Find the set of values of $x$ for which the sum to infinity exists.

A pendulum swings through an arc of $40$ cm on its first swing. Each later swing is $90\%$ of the length of the swing before.

Calculate the total length of arc swung if the motion continues indefinitely.

Consider the infinite geometric series $0.072+0.00072+0.0000072+\dots$

Find the exact value of the sum to infinity, giving your answer as a fraction in its lowest terms.

The function $h$ is defined by $h(x)=4-\frac{x}{3}$. It is given that $h^n(x)={\left(-\frac{1}{3}\right)}^{n}x+3\left(1-{\left(-\frac{1}{3}\right)}^n\right)$.

As $n$ grows without bound the graphs of $y=h^n(x)$ settle onto a horizontal line. Determine its equation.

The first three terms of an infinite geometric series are $2\ln x$, $\ln x$ and $\frac{1}{2}\ln x$, where $x>1$.

Given that the sum to infinity is $12$, find the exact value of $x$.

A pendulum bob swings either side of its rest position O. Its greatest distance from O on the first swing is 3.2 cm, and each later greatest distance is 0.36 times the one before. On every swing it goes out to that distance and comes back to O.

Calculate the total distance the bob travels if it keeps swinging in this way forever.