Question

Find the common difference and write the next four terms of the given arithmetic progression: 1, -2, -5, -8, …

Answer 1

Hint: In these types of question use the given terms to find the common difference with the help of the formula i.e. $d = {a_{n + 1}} - {a_n}$ use numbers of terms accordingly in the formula and find the way to approach toward the solution.

Complete step-by-step answer:
According to the given information A.P is 1, -2, -5, -8, …
So we have \[{a_1}\] = 1, \[{a_2}\] = -2, ${a_3}$ = -5, \[{a_4}\] = -8
To find the common difference between the terms we will use the equation which is given by $d = {a_{n + 1}} - {a_n}$
In the above equation let using the value of n as 1
$d = {a_{1 + 1}} - {a_1}$
$ \Rightarrow $$d = {a_2} - {a_1}$ (Equation 1)
Substituting the given values of \[{a_2}\] and \[{a_1}\] in the equation 1
$d = - 2 - 1$
$ \Rightarrow $$d = - 3$
Now using the formula of nth term of A.P i.e. ${a_n} = {a_1} + \left( {n - 1} \right)d$ to find the next four terms in the given A.P
Substituting the given values in the formula
${a_n} = 1 + \left( {n - 1} \right)\left( { - 3} \right)$
$ \Rightarrow $${a_n} = 4 - 3n$ (Equation 2)
So we already have the first 4 terms of A.P and we have to find the next 4 terms
Since the \[{a_4}\] is last given term of A.P
Therefore the for the 5th term n = 5
Substituting the value of n in equation 2
${a_5} = 4 - \left( {3 \times 5} \right)$
$ \Rightarrow $${a_5} = - 11$
For the 6th term n = 6
Substituting the value of n in equation 2
\[{a_6} = 4 - \left( {3 \times 6} \right)\]
$ \Rightarrow $\[{a_6} = - 14\]
Now for the 7th term of A.P n = 7
Substituting the value of n in equation 2
${a_7} = 4 - \left( {3 \times 7} \right)$
$ \Rightarrow $${a_7} = - 17$
For the 8th term of the given A.P n = 8
Substituting the value of n in equation 2
${a_8} = 4 - \left( {3 \times 8} \right)$
$ \Rightarrow $${a_8} = - 20$
So the next four terms of the given A.P are -11, -14, -17, and -20.

Note: The term A.P that was the concept behind the above problem A.P stands for Arithmetic progression which can be explained as the sequence of numbers with common difference between two consecutive terms for a sequence which consists of finite numbers is named as finite arithmetic progression.