Question

Please... ?

The first term of an infinite geometric progression is 5 and the sum of its terms is 20. What is the common ratio of the progression?

Answer 1

Answer:

• The common ratio of the progression is 3/4

Explanation:

A geometric progression is a sequence of terms in which the consecutive terms have a constant ratio; thus, each term is equal to the previous one multiplied by a constant value:

$First\ term=a_1\\\\ Second\ term=a_2=a_1\times r\\\\ Third\ term=a_3=a_2\times r=a_1\times r^2\\\\n_{th}\ term=a_n=a_{n-1}\times r=a_1\times r^{n-1}$

A infinite geometric progression may have a finite sum. When the absolute value of the ratio is less than 1, the sum of the infinite geometric progression has a finite value equal to:

• $S_{\infty}=\frac{a_1}{1-r}$

Thus, the information given translates to:

$a_1=5\\ \\ S_{\infty}=20=\frac{5}{1-r}$

Now you can solve for the constant ratio, r:

$1-r=\frac{5}{20}\\ \\ r=1-\frac{5}{20}\\ \\ r=\frac{15}{20}\\  \\ r=3/4$