7.5 Infinite geometric series practice 2

Evaluate [math]\displaystyle{\sum^\infty_{n=1}\left(\dfrac{3}{4}\right)^n}[/math]

A super ball dropped from a height of 30 yards rebounded on each bounce [math]\dfrac{3}{8}[/math] of the height from which it fell. How far (in yards) did the vall travel before coming to rest?

Timmy created an infinite amout of circles all with a special property. Starting with the first circle of radius 1, which successive circle that Timmy created has a radius numerically equal to the area enclosed by the previous circle. What is the enclosed area of the [math]11^{th}[/math] circle that Timmy created?

Consider the infinite geometric series [math]1+\dfrac{2x}{3}+\left(\dfrac{2x}{3}\right)^2+\left(\dfrac{2x}{3}\right)^3+\left(\dfrac{2x}{3}\right)^4\cdots[/math]

(a) For what values of [math]x[/math] does the series converge?

(b) Find the sum of the series if [math]x=1.2[/math]

Calculate [math]\displaystyle{\sum^{2013}_{n=1}\log_{2014}{\dfrac{n+1}{n}}}[/math]

The sum to infinity of geometric progression is 3. When the terms of this geometric progression are squared a new geometric progression is obtained whose sum to infinity is 1.8. Find the first term and the common ratio of the new sequence.

Write [math]3.8\overline{24}[/math] as a fraction [math]\dfrac{m}{n}[/math] where [math]m[/math] and [math]n[/math] are relative prime positive integers.

Consider the infinite geometric series [math]1+(\dfrac {3x}{5})+(\dfrac {3x}{5})^2+(\dfrac {3x}{5})^3+\cdots[/math]

(a) Find the conditions of [math]x[/math] such that this infinite series is convergent.

(b) Find the sum of the series if [math]x = 1.5[/math].

If [math] \dfrac{1}{2}+ \dfrac{\sqrt{2}}{8}+ \dfrac{1}{16}+ \dfrac{\sqrt{2}}{64}+ \dfrac{1}{128}+\cdots=a+b\sqrt{2}[/math] where [math]a, b\in\mathbb{Q}[/math], what is the value of [math]a+b[/math]?

The sum of an inifinite geometric series with commmon ratio [math]r[/math] such that [math] |r|<1[/math] is 15 and the sum if the squares of the terms of this series is 45. What is the value of the common ratio?

Find the sum of each expession.

(a) [math]\displaystyle{\sum^6_{i=1}(2i-6)}=[/math]

(b) [math]\displaystyle{\sum^7_{k=3}(k+4)}=[/math]

(c) [math]\displaystyle{\sum^\infty_{k=1}\left(16\left(\displaystyle{-\dfrac{1}{3}}\right)^{k-1}\right)}=[/math]