Question

In an A.P, if \[{n^{th}}\] term is $7 - 4n$, Find the common difference?

Answer 1

Hint: The general form of an Arithmetic Progression is \[a\] , \[a + d\] , \[a + 2d\] , \[a + 3d\] and so on. The formula for \[\;{n^{th}}\;\] term of an AP series is \[{T_n} = {\text{a}} + \left( {n - 1} \right)d\] , where \[{T_n}\; = {n^{th}}\;term\] , a = first term and \[d = common\,difference = {T_n}\; - {\text{ }}{T_{n - 1}}\] . We should take care that the coefficient of d is always$1$less than the term number.

Complete step-by-step answer:
An arithmetic progression is a set of numbers with a common difference between any two subsequent numbers is always constant.
We have given \[\;{n^{th}}\;\] term, \[7 - 4n\]
And we know that a common difference is the difference of two consecutive terms in an A.P.
So,
 \[{T_n} = 7 - 4n{\text{ }}and{\text{ }}{T_{n - 1}} = 7 - 4\left( {n - 1} \right)\]
Common difference, \[d = {T_n}\; - {\text{ }}{T_{n - 1}}\]
 \[d = 7 - 4n - \left\{ {7 - 4\left( {n - 1} \right)} \right\}\]
By further solving
 \[d = 7 - 4n - \left\{ {7 - 4n + 4} \right\}\]
 \[d = 7 - 4n - 7 + 4n - 4\]
By cancelling the positive and negative we get,
 \[d = -4\]
So, the common difference is -4
So, the correct answer is “-4”.

Note: An arithmetic progression is a set of numbers with a common difference between any two subsequent numbers (A.P.). A.P.'s example is 3,6,9…
Sum of n terms in AP, \[S = \dfrac{n}{2}\left[ {2a + \left( {n - 1} \right) \times d} \right] \] \[\] ,the sum of AP when first and last terms are given \[S = \dfrac{n}{2}\left[ {a + l} \right] \] , Where, first term as ‘a’ , n is number of elements and last term as ‘l’.
A geometric progression is a sequence where every term bears a constant ratio to its preceding term. Geometric progression is a special type of sequence.