Question

How do you write the first five terms of the geometric sequence a = 8 and r = 5?

Answer 1

Hint: In this question, we are given the first term 'a' and the common ratio r of a geometric sequence and we need to find the first five terms of the sequence. For this we will use the general formula of calculating the general term of a sequence. The formula is given by ${{a}_{n}}=a{{r}^{n-1}}$ where we are finding ${{n}^{th}}$ term of the geometric sequence having first term as a and common ratio as r. We will put values of n as 1, 2, 3, 4, 5 to calculate the first five terms of the geometric progression.

Complete step-by-step answer:
Here we are given a geometric sequence whose first term is a which is equal to 8 and the common ratio is r which is equal to 5. We need to find the first five terms of this geometric sequence. For this, let us use the formula of calculating the general term of a geometric sequence.
The formula for finding ${{n}^{th}}$ term in the geometric sequence is given as ${{a}_{n}}=a{{r}^{n-1}}$ where a is the first term of the sequence, r is the common ratio and ${{a}_{n}}$ is the required ${{n}^{th}}$ term.
Here we want the first five terms so let us put n = 1, 2, 3, 4, 5 separately to find the required terms.
We have a = 8 and r = 5 and ${{a}_{n}}=a{{r}^{n-1}}$.
Let us put n = 1 to find the first term i.e. ${{a}_{1}}=8{{\left( 5 \right)}^{1-1}}\Rightarrow {{a}_{1}}=8{{\left( 5 \right)}^{0}}$.
We know that ${{x}^{0}}=1$ where x is any integer so, ${{a}_{1}}=8$ which is the first term.
Now let us put n = 2, to find the second term i.e. ${{a}_{2}}=8{{\left( 5 \right)}^{2-1}}\Rightarrow {{a}_{2}}=8{{\left( 5 \right)}^{1}}$.
We know that 8 times 5 is 40 so we get ${{a}_{2}}=40$.
So the second term of the sequence is 40.
Now let us put n = 3 to find the third term i.e. ${{a}_{3}}=8{{\left( 5 \right)}^{3-1}}\Rightarrow {{a}_{3}}=8{{\left( 5 \right)}^{2}}$.
We know that, ${{5}^{2}}=5\times 5$ which is equal to 25 so we get ${{a}_{3}}=8\times 25\Rightarrow {{a}_{3}}=200$.
So the third term of the sequence is 200.
Now let us put n = 4 to find the fourth term i.e. ${{a}_{4}}=8{{\left( 5 \right)}^{4-1}}\Rightarrow {{a}_{4}}=8{{\left( 5 \right)}^{3}}$.
We know that ${{5}^{3}}=5\times 5\times 5$ which is equal to 125 so we get ${{a}_{4}}=8\times 125\Rightarrow {{a}_{4}}=1000$.
So the fourth term of the sequence is 1000.
Now let us put n = 5 to get the fifth term of the sequence so we get ${{a}_{5}}=8{{\left( 5 \right)}^{5-1}}\Rightarrow {{a}_{5}}=8{{\left( 5 \right)}^{4}}$.
We know that ${{5}^{4}}=5\times 5\times 5\times 5$ which is equal to 625 so we get ${{a}_{5}}=8\times 625\Rightarrow {{a}_{5}}=5000$.
So the fifth term of the sequence is 5000.
Hence the first five terms of the given geometric sequence are 8, 40, 200, 1000, 5000.

Note: Students should carefully calculate all the terms by proper multiplication. Note that students need not calculate the first term as it is already given. They can start from n = 2 also. Students can get confused between ${{a}_{n}}$ and n. Here ${{a}_{n}}$ is the ${{n}^{th}}$ term of the sequence.