Question

Given an A.P whose terms are all positive integers. The sum of its first nine terms is greater than 200 and less than 220. If the second term in it is 12, then 4th is:
A. 8
B. 16
C. 20
D. 24

Answer 1

Hint: This question can be done by easily understanding the concepts of arithmetic progression and inequality which are mentioned below: -
*$n^{th}$ term of an A.P (arithmetic progression) is given by ${T_n} = a + (n - 1)d$ where a= first term of the sequence and d=common difference which is given by $d = {T_n} - {T_{n - 1}}$
*Sum of n terms is given by ${S_n} = \dfrac{n}{2}[2a + (n - 1)d]$

Complete step-by-step answer:
First of all we will apply ${T_n} = a + (n - 1)d$ formula so that we can obtain one equation. Now in question we are given a second term so we will apply values in the above equation for the second term.
$ \Rightarrow {T_2} = a + (2 - 1)d$ (Here n=2)
$ \Rightarrow 12 = a + d$ (As second term is given i.e. 12)
 $ \Rightarrow a = 12 - d$ .................equation (1)
Now it is given in question that the sum of its first nine terms is greater than 200 and less than 220. For solving we required one more equation so we will obtain second equation from this condition and for that first we will find sum of nine terms by applying formula ${S_n} = \dfrac{n}{2}[2a + (n - 1)d]$
Sum of (n=9) terms = ${S_9} = \dfrac{9}{2}[2a + (9 - 1)d]$
$ \Rightarrow {S_9} = \dfrac{9}{2}[2a + 8d]$
$ \Rightarrow {S_9} = 9[a + 4d]$ (Taking 2 common and then cancelling from the denominator)
Now we will put the value of ‘a’ from equation 1 in the above sum of n terms equation.
$ \Rightarrow {S_9} = 9[12 - d + 4d]$
$ \Rightarrow {S_9} = 9[12 + 3d]$
Now we will apply the inequality condition that is given in the question.
$ \Rightarrow 200 < {S_9} < 220$ (Given:- Sum of its first nine terms is greater than 200 and less than 220)
Now we will put ${S_9} = 9[12 + 3d]$ in the inequality equation.
$ \Rightarrow 200 < 9[12 + 3d] < 220$
$ \Rightarrow 200 < 108 + 27d < 220$ (Opening brackets and multiplying terms)
Now we will subtract 108 from all terms of the equation.
 $ \Rightarrow 200 - 108 < 108 + 27d - 108 < 220 - 108$
$ \Rightarrow 92 < 27d < 112$
Now dividing entire equation by 27
$ \Rightarrow \dfrac{{92}}{{27}} < \dfrac{{27d}}{{27}} < \dfrac{{112}}{{27}}$
$ \Rightarrow 3.40 < d < 4.148$
So, we will take middle value i.e. 4
Therefore common difference (d) = 4 and first term (a) = 12-d = 12-4 = 8
Now applying ${T_n} = a + (n - 1)d$ for fourth term
$ \Rightarrow {T_4} = 8 + (4 - 1)4$
$ \Rightarrow {T_4} = 8 + 12 = 20$
Therefore the fourth term of A.P is 20.

So, the correct answer is “Option C”.

Note: While solving inequality we must pay attention to the direction of the inequality. In which direction an arrow should point plays a vital role. Closed side is for lesser quantity while the open side is for greater quantity.
Things which do not affect the inequality:-
1. Add or subtract a number from both the sides.
2. Multiply or divide both sides from a positive number.
3. Simplification can be done within sides.
Things which do not affect the inequality:-
1. Multiply or divide both sides from a negative number.
2. Swapping left or right hand sides.