Question

Find the common difference and write the next four terms of each of the following arithmetic progression:
-1,$\dfrac{-5}{6}$,$\dfrac{-2}{3}$,…….

Answer 1

Hint: We will be using the concept of sequence and series and Arithmetic progression to solve the question. An arithmetic progression (AP) or arithmetic sequence is a series such that the difference between the consecutive terms is constant. The difference here means the second minus the first. For instance, sequence 5, 7, 9, 11, 13, 15, . . . is an arithmetic progression with a common difference of 2.

Complete step by step answer:

Now we have been given an arithmetic series as follows:
-1,$\dfrac{-5}{6}$,$\dfrac{-2}{3}$,…….
We have to find the common difference and next four terms in the series of the AP. To find the common difference and the next four terms of the arithmetic progression we have to find the first term of the series and common difference.
First term of the series =a = -1
Common difference of the series=d= ($\dfrac{-5}{6}$- (-1)) =$\dfrac{1}{6}$
Let ${{a}_{1}}$,${{a}_{2}}$,${{a}_{3}}$,${{a}_{4}}$ be the next four terms in the series of the AP.
We know that the next term in the series can be found by adding a common difference in the previous term. Therefore,
${{a}_{1}}$=$\dfrac{-2}{3}$+$\dfrac{1}{6}$=$\dfrac{-1}{2}$
${{a}_{2}}$=${{a}_{1}}$+$\dfrac{1}{6}$=$\dfrac{-1}{3}$
${{a}_{3}}$=${{a}_{2}}$+$\dfrac{1}{6}$=$\dfrac{-1}{6}$
${{a}_{4}}$=${{a}_{3}}$+$\dfrac{1}{6}$=$0$
Therefore the common difference of the AP is d=$\dfrac{1}{6}$ and the next four terms in the AP are $\dfrac{-1}{2}$,$\dfrac{-1}{3}$,$\dfrac{-1}{6}$,0.
Note: There is an alternative method by using the standard series formula. For example, the terms can be found by using the relation of nth term with common difference (d) and first term (a) as
L (nth terms) = a+(n-1) d
Using this relation and substituting the values of n as 4,5,6,7 we get the 4th,5th,6th,7th terms as
$\dfrac{-1}{2}$,$\dfrac{-1}{3}$,$\dfrac{-1}{6}$,0.