Question

Which one of the following is the correct answer?
A. $ {Q_1},{\text{ }}{Q_2},\,{Q_3},..... $ are in A.P. with common difference $ 5 $
B. $ {Q_1},{\text{ }}{Q_2},\,{Q_3},..... $ are in A.P. with common difference $ 6 $
C. $ {Q_1},{\text{ }}{Q_2},\,{Q_3},..... $ are in A.P. with common difference $ 11 $
D. $ {Q_1} = {Q_2} = \,{Q_3} = ..... $

Answer 1

Hint: An Arithmetic Progression (AP) is the sequence of numbers in which the difference of two successive numbers is always constant.
The standard formula for Arithmetic Progression is – $ {a_n} = a + (n - 1)d $
Where $ {a_n} = $ nth term in the AP
 $ a = $ First term of AP
 $ d = $ Common difference in the series
 $ n = $ Number of terms in the AP
Here first of all we will assume the $ {T_r}{\text{ and }}{T_{r + 1}} $ where $ {Q_r} = {T_{r + 1}} - {T_r} $

Complete step-by-step answer:
Let us assume that –
 $ {T_r} = 3{r^2} + 2r - 1 $ ... (a)
And $ {T_{r + 1}} = 3{(r +1)^2} + 2(r + 1) - 1 $ .... (b)
Also, $ {Q_r} = {T_{r + 1}} - {T_r} $
Place the values from equation (a) and (b) in the above equation.
 $ {Q_r} = [3{(r + 1)^2} + 2(r + 1) - 1] - [3{r^2} + 2r - 1] $
Simplify the above expression, apply the formula for the difference of two terms whole square. $ {(a + b)^2} = {a^2} +2ab + {b^2} $
 $ {Q_r} = [3({r^2} + 2r + 1) + 2(r + 1) - 1] - [3{r^2} + 2r - 1] $
Remember when you open the bracket and there is a negative sign outside it, all the terms inside the bracket changes. Positive terms become negative and vice-versa.
 $ {Q_r} = [3{r^2} + 6r + 3 + 2r + 2 - 1 - 3{r^2} - 2r + 1] $
Make the pair of like terms –
 $ {Q_r} = [\underline {3{r^2} - 3{r^2}} \underline { + 6r - 2r + 2r} + \underline {3 + 2 - 1 + 1} ] $
Like terms with equal values and opposite signs cancel each other.
 $ {Q_r} = [ + 6r + 5] $ ..... (c)
Similarly, $ {Q_{r + 1}} = [6(r + 1) + 5] $ ..... (d)
Now, the common difference between the above two equations $ = {Q_{r + 1}} - {Q_r} $
Place the values –
The common difference $ = [6(r + 1) + 5] - [6r + 5] $
Open the brackets and simplify –
The common difference $ = [6r + 6 + 5 - 6r - 5] $
Like terms with equal values and opposite signs cancel each other.
The common difference $ = 6 $
So, the correct answer is “Option B”.

Note: Know the difference between the arithmetic and geometric progression. In arithmetic progression, the difference between the numbers are constant in the series whereas, the geometric progression is the sequence in which the succeeding element is obtained by multiplying the preceding number by the constant and the same continues for the series. The ratio between the two remains the same.