Finding the nth term given the common ratio and the first Term

We will discuss here of finding the nth term given the common ratio and the first Term. Here the nth term means any indicated term. For this we will use the following formula as
$a_{n} = a r^{n - 1}$
where 'a' is the first term and 'r' is the common ratio.

Examples on finding the nth term given the common ratio and the first Term

Example : 1 Find the 9th term of the G.P. 1,4,16,64,...
Solution : The sequence is 1,4,16,64,...
Here, first term = a = 1
common ratio = r = $\frac{4}{1} = \frac{16}{4}$ = 4
The general term of geometric progression is given by
$a_{n} = ar^{n - 1}$
Since we have to find the 9th term,
n = 9
∴ $a_{9} = 1 \times 4^{9 - 1}$
= 1 $\times 4^{8}$
∴ $a_{9} = 4^{8}$

Example : 2 Find the 10th term of the G.P. 64,32,16,8,...
Solution : The sequence is 64,32,16,8,...
Here, first term = a = 64
common ratio = r = $\frac{32}{64} = \frac{16}{32} =\frac{1}{2}$
The general term of geometric progression is given by
$a_{n} = ar^{n - 1}$
Since we have to find the 10th term,
n = 10
∴ $a_{10} = 64 \times \frac{1}{2^{10 - 1}}$

= 64 $\times \frac{1}{2^{9}}$

= $2^{6}\times \frac{1}{2^{9}}$

= $\frac{1}{2^{9 - 6}}$

= $ \frac{1}{2^{3}}$

∴ $a_{10} = \frac{1}{8}$

Example 3: The first term and the common ratio of a geometric sequence are 0.8 and -5 respectively. Find the 5th term.
Solution : Here, first term = a = 0.8
common ratio = r = -5
The general term of geometric progression is given by
$a_{n} = ar^{n - 1}$
Since we have to find the 5th term,
n = 5
∴ $a_{5} = 0.8 \times (-5)^{5 - 1}$
= 0.8 $\times (-5)^{4}$
∴ $a_{5}$ = 500

Example 4: Find the 8th term, if the first term and the common ratio of a geometric sequence are 45 and 0.2 respectively.
Solution : Here, first term = a = 45
common ratio = r = 0.2
The general term of geometric progression is given by
$a_{n} = ar^{n - 1}$
Since we have to find the 8th term,
n = 8
∴ $a_{8} = 45 \times (0.2)^{8 - 1}$
= 45 $\times (0.2)^{7}$
∴ $a_{8}$ = 0.000576


11th grade math

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