Question

How do you find the next three terms of the arithmetic sequence $ 3.1,4.1,5.1,6.1,........ $ ?

Answer 1

Hint: In order to determine the next three terms of the given arithmetic sequence, find out the first term $ a $ which will be equal to $ 3.1 $ and the common difference by subtracting any two consecutive terms from the sequence. Now put these values along with $ n = 5 $ , $ n = 6 $ and $ n = 7 $ (because four terms are given previously) in the formula of nth term of AP, $ {a_n} = a + (n - 1)d $ to obtain the desired result i.e the next three terms.

Complete step by step solution:
Clearly, the given sequence is an Arithmetic Progression (A.P.).
As we know the nth term of an A.P. is $ {a_n} = a + (n - 1)d $
where $ a $ is the first term, $ d $ is the common constant difference
So, the $ 5th $ term of the A.P. will be $ {a_5} $
In our sequence first term $ a = 3.1 $
Difference ( $ d $ ) can be calculated by subtracting any two consecutive terms, we get
 $ d = 4.1 - 3.1 = 1 $
According to the question we have to find the value of the next three terms which would be $ n = 5 $ , $ n = 6 $ and $ n = 7 $ terms.
Now putting the values of $ n,a\,and\,d $ in the $ nth $ term of A.P. we get
 $
  {a_n} = a + (n - 1)d \\
  {a_5} = 3.1 + (5 - 1)(1) \\
  {a_5} = 3.1 + (4)(1) \\
  {a_5} = 3.1 + 4 \\
  {a_5} = 7.1 \;
  $
Therefore, 5th term $ \left( {{a_5}} \right) $ of the given arithmetic sequence is equal to $ 7.1 $ .
Now, for next term that is $ n = 6 $ , put the values of $ n,a\,and\,d $ in the $ nth $ term of A.P. we get:
 $
  {a_n} = a + (n - 1)d \\
  {a_6} = 3.1 + (6 - 1)(1) \\
  {a_6} = 3.1 + (5)(1) \\
  {a_6} = 3.1 + 5 \\
  {a_6} = 8.1 \;
  $
Therefore, the 6th term $ \left( {{a_6}} \right) $ of the given arithmetic sequence is equal to $ 8.1 $ .
Now, for next term that is $ n = 7 $ , put the values of $ n, a\,and\, d $ in the $ nth $ term of A.P. we get:
 $
  {a_n} = a + (n - 1)d \\
  {a_7} = 3.1 + (7 - 1)(1) \\
  {a_7} = 3.1 + (6)(1) \\
  {a_7} = 3.1 + 6 \\
  {a_7} = 9.1 \;
   $
Therefore, the 7th term $ \left( {{a_7}} \right) $ of the given arithmetic sequence is equal to $ 9.1 $ .
Hence, the next three terms of the arithmetic sequence $ 3.1,4.1,5.1,6.1,........ $ are $ 7.1 $ , $ 8.1 $ and $ 9.1 $ .
So, the correct answer is “ $ 7.1 $ , $ 8.1 $ and $ 9.1 $ ”.

Note: 1.Don’t forgot to cross-check your answer.
2.The difference between any two consecutive terms in an A.P. is always the same and if it is not the same, then the given series is not an A.P.
3. $ (n - 1) $ is nothing but the position of term in the sequence.