Question

Write the next term of A.P. $ \sqrt 8 ,\sqrt {18} ,\sqrt {32} ,.... $

Answer 1

Hint: The given question deals with the concept of arithmetic progression. In order to solve this question, we should have prior knowledge of arithmetic progression. An interesting fact- the common difference of an arithmetic progression remains the same throughout the series.

Complete step by step solution:
A list of numbers in which a new number is obtained by adding a fixed number to the preceding term except the first term is known as an arithmetic progression. The first term of an arithmetic progression is denoted by $ {a_1} $ whereas the common difference is denoted by $ d $ . The $ {n^{th}} $ term $ {a_n} $ of an AP with first term $ a $ and common difference $ d $ is denoted by:
 $ \Rightarrow {a_n} = a + \left( {n - 1} \right)d $
Given is A.P. $ \sqrt 8 ,\sqrt {18} ,\sqrt {32} ,.... $
The given arithmetic progression can be rewritten as:
 $ 2\sqrt 2 ,3\sqrt 2 ,4\sqrt 2 ,..... $
Let us now find the common difference $ d $ ,
The common difference ( $ {d_1} $ ) is $ 3\sqrt 2 - 2\sqrt 2 = \sqrt 2 $
The common difference ( $ {d_2} $ ) is $ 4\sqrt 2 - 3\sqrt 2 = \sqrt 2 $
As we can clearly see that the common difference is the same, this even proves that the common difference remains the same throughout the series. So, $ d = \sqrt 2 $
In order to calculate the next term, we will add the preceding term to the common difference. Doing so, we get
 $
   \Rightarrow 4\sqrt 2 + \sqrt 2 \\
   \Rightarrow 5\sqrt 2 \;
  $
The series now becomes, $ 2\sqrt 2 ,3\sqrt 2 ,4\sqrt 2 ,5\sqrt 2 $
This can be rewritten as:
 $ \sqrt 8 ,\sqrt {18} ,\sqrt {32} ,\sqrt {50} $
Therefore, the next term of the A.P. is $ \sqrt {50} $ .
So, the correct answer is “ $ \sqrt {50} $ ”.

Note: Here, in this question it was clearly mentioned that it is an A.P. Students should be able to identify the given series whether it is an arithmetic progression or geometric progression or harmonic progression. Students should remember the fact that common difference remains the same throughout the series.