Question

How do you express the ${n^{th}}$ term of the geometric sequence ${a_1} = 4,r = \dfrac{1}{2},n = 10$?

Answer 1

According to given in the question we have to determine the expression for the ${n^{th}}$ term of the geometric sequence${a_1} = 4,r = \dfrac{1}{2}, n = 10$. So, to determine the ${n^{th}}$ term of the geometric sequence first of all we have to understand about the geometric sequence which is as explained below:
Geometric sequence: Geometric sequence is also known as the geometric progression which is a sequence of non-zero terms where each term after the first term is found by multiplying the previous one by a fixed, non-zero number which is known as common ratio and it is represented by the symbol r.
Now, we have to determine the ${n^{th}}$ term of the geometric sequence which can be determined with the help of the formula of finding the ${n^{th}}$ term of the geometric sequence which is as mentioned below:

Formula used:
$\Rightarrow {a_n} = {a_1}{(r)^{n - 1}}.................(A)$
Where, ${a_n}$is the ${n^{th}}$term of the geometric sequence and r is the common ratio and n is the given number of terms.
Now, we have to substitute all the values in the formula (A) which is as mentioned above.
Now, on solving all the values we can easily determine the required ${n^{th}}$term of the geometric sequence.

Complete step by step solution:
Step 1: First of all we have to determine the ${n^{th}}$term of the geometric sequence which can be determined with the help of the formula (A) of finding the ${n^{th}}$term of the geometric sequence which is as mentioned in the solution hint. Hence,
$\Rightarrow {a_1} = 4$which is the first term of the geometric sequence,
$r = \dfrac{1}{2}$is the common ratio of the geometric sequence and,
$n = 10$is the number of terms we have to determine for the sequence.
Step 2: Now, we have to substitute all the values in the formula (A) which is as mentioned above. Hence,
$\Rightarrow {a_{10}} = 4{\left( {\dfrac{1}{2}} \right)^{10 - 1}}$
Step 3: Now, on solving all the values as obtained in the solution step 2 we can easily determine the required ${n^{th}}$term of the geometric sequence. Hence,
$\Rightarrow {a_{10}} = 4{\left( {\dfrac{1}{2}} \right)^9} \\\ \Rightarrow {a_{10}} = 4 \times \dfrac{1}{{512}} \\\ \Rightarrow {a_{10}} = \dfrac{1}{{128}} \\\$

Hence, with the help of the formula (A) we have determined the required ${n^{th}}$term of the geometric sequence which is $\Rightarrow {a_{10}} = 4 \times \dfrac{1}{{512}}$.

Note:

1. Geometric progression is a sequence of non-zero terms where each term after the first term is found by multiplying the previous one by a fixed, non-zero number which is known as common ratio and it is represented by the symbol r.
2. The distinction between a progression and a series is that a progression is a sequence, whereas series is a sum.