problem solving of geometric sequence

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Practice Problems on Geometric Series

A  geometric series  is a type of infinite series formed by summing the terms of a geometric sequence. A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the  common ratio .

The general form of a geometric series can be expressed as:

S = a + ar + ar^2 + ar^3 + ar^4 + \ldots

• S is the sum of the series.
• a is the first term.
• r is the common ratio.

Sample Questions of Geometric Series

Question 1: What is the Geometric mean 2, 4, 8?

Solution:  

According to the formula,  =\sqrt [3]{(2)(4)(8)}\\=4

Question 2: Find the first term and common factor in the following Geometric Progression:

4, 8, 16, 32, 64, . . .

Here, It is clear that the first term is 4, a=4 We obtain common Ratio by dividing 1st term from 2nd: r = 8/4 = 2

Question 3: Find the 8 th and the n th term for the G.P: 3, 9, 27, 81, . . .

Solution: 

Put n=8 for 8 th term in the formula: ar n-1 For the G.P : 3, 9, 27, 81 . . . First term (a) = 3 Common Ratio (r) = 9/3 = 3 8 th term = 3(3) 8-1 = 3(3) 7 = 6561 N th = 3(3) n-1 = 3(3) n (3) -1 = 3 n

Question 4: For the G.P. : 2, 8, 32, . . . which term will give the value 131073?

Assume that the value 131073 is the N th term, a = 2, r = 8/2 = 4 N th term (a n ) = 2(4) n-1 = 131073 4 n-1 = 131073/2 = 65536 4 n-1 = 65536 = 4 8 Equating the Powers since the base is same: n-1 = 8 n = 9

Question 5: Find the sum up to 5 th and N th term of the series:  1, \frac{1}{2},\frac{1}{4},\frac{1}{8}...

a= 1, r = 1/2 Sum of N terms for the G.P,  {S_n =\frac{a(1-r^n)}{1-r}}                                                  = \frac{1(1-(\frac{1}{2})^n)}{1-\frac{1}{2}}   = 2 (1-(\frac{1}{2})^n) Sum of first 5 terms ⇒ a 5 =  2 ( 1-(\frac{1}{2})^5) =  2 ( 1-(\frac{1}{32})) =  (\frac{31}{16})

Question 6: Find the Sum of the Infinite G.P: 0.5, 1, 2, 4, 8, ...

Formula for the Sum of Infinite G.P:  \frac{a}{1-r}  ; r≠0 a = 0.5, r = 2 S ∞ = (0.5)/(1-2) = 0.5/(-1)= -0.5

Question 7: Find the sum of the Series: 5, 55, 555, 5555,... n terms

Solution : 

The given Series is not in G.P but it can easily be converted into a G.P with some simple modifications. Taking 5 common from the series: 5 (1, 11, 111, 1111,... n terms) Dividing and Multiplying with 9:  \frac{5}{9}(9+ 99+ 999+...n terms) ⇒  \frac{5}{9}[((10+(10)^2+(10)^3+...n terms)-(1+1+1+...n terms)] ⇒ \frac{5}{9}[(\frac{10((10)^n-1)}{10-1})-(n)] ⇒  \frac{5}{9}[(\frac{10((10)^n-1)}{9})-(n)]

Worksheet: Geometric Series

You can download this free worksheet with answer key from follows:

• Sequence and Series
• Geometric Series
• Sum of Infinite Geometric Series
• Arithmetic Series

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