Question

The 7th term of the sequence\[\sqrt 2 ,\sqrt {10} ,5\sqrt 2 ...\] is
1) \[125\sqrt {10} \]
2) \[25\sqrt 2 \]
3) \[125\]
4) \[125\sqrt 2 \]

Answer 1

Hint: This is a question on elementary geometric progression. It involves basic concepts like the common ratio r and the nth term of the geometric progression. The formula used to solve this question is:
The common ratio of the geometric progression, \[r = \dfrac{{{a_n}}}{{{a_{n - 1}}}}\]
The nth term of the geometric progression, \[{a_n} = a{r^{n - 1}}\]

Complete step-by-step answer:
Let us first try to find out which type of progression this particular sequence follows,
\[ \Rightarrow \sqrt 2 ,\sqrt {10} ,5\sqrt 2 ...\]
Let us first check for arithmetic progression,
Finding the difference between the first and the second term
\[ \Rightarrow {d_1} = \sqrt {10} - \sqrt 2 \]
And then, finding the difference between the second and the third term
\[ \Rightarrow {d_2} = 5\sqrt 2 - \sqrt {10} \]
Therefore, \[{d_1} \ne {d_2}\].
The two differences are not equal, and hence the above sequence is not an arithmetic progression.
Now, let us check the sequence for geometric progression,
Now, finding the common ratio for the first and second term
\[ \Rightarrow {r_1} = \dfrac{{\sqrt {10} }}{{\sqrt 2 }}\]
\[ \Rightarrow {r_1} = \sqrt 5 \]
Similarly, finding the common ratio for the second and third term
\[ \Rightarrow {r_2} = \dfrac{{5\sqrt 2 }}{{\sqrt {10} }}\]
\[ \Rightarrow {r_2} = \sqrt 5 \]
\[ \Rightarrow {r_1} = {r_2}\]
The two common ratios are equal, and hence, the above progression is geometric.
Now, the first term a and the common ratio r is given by
\[ \Rightarrow a = \sqrt 2 \]
\[ \Rightarrow r = \sqrt 5 \]
Now, the 7th term of the sequence is given by
\[ \Rightarrow {a_n} = a{r^{n - 1}}\]
Now, putting the value of n=7 to find the value of the 7th term
\[ \Rightarrow {a_7} = a{r^{7 - 1}}\]
\[ \Rightarrow {a_7} = a{r^6}\]
Now, replacing a and r from the above equations, we get,
\[ \Rightarrow {a_7} = \sqrt 2 {(\sqrt 5 )^6}\]
\[ \Rightarrow {a_7} = \sqrt 2 \times {5^3}\]
And finally, we get the value of the 7th term as shown below.
\[ \Rightarrow {a_7} = 125\sqrt 2 \]
Therefore, option(4) is the correct answer.
So, the correct answer is “Option 4”.

Note: This question requires one to observe a sequence and figure out a common pattern. After finding the pattern, apply suitable formulas to solve the question. Do not commit calculation mistakes, and be sure of the final answer.