If the sums of n terms of two arithmetic progressions are in the ration $$\frac{{3n + 5}}{{5n + 7}},$$ then their nth terms are in the ration
A. $$\frac{{3n - 1}}{{5n - 1}}$$
B. $$\frac{{3n + 1}}{{5n + 1}}$$
C. $$\frac{{5n + 1}}{{3n + 1}}$$
D. $$\frac{{5n - 1}}{{3n - 1}}$$
Answer: Option B
Solution (By Examveda Team)
In first A.P. let its first term be a1 and common difference d1and in second A.P., first term be a2 and common difference d2, then
$$\eqalign{ & \frac{{{S_n}}}{{{S_n}}} = \frac{{\frac{n}{2}\left[ {2{a_1} + \left( {n - 1} \right){d_1}} \right]}}{{\frac{n}{2}\left[ {2{a_2} + \left( {n - 1} \right){d_2}} \right]}} \cr & \,\,\,\,\,\,\,\,\,\,\, = \frac{{2{a_1} + \left( {n - 1} \right){d_1}}}{{2{a_2} + \left( {n - 1} \right){d_2}}} \cr} $$
$$\therefore \frac{{2{a_1} + \left( {n - 1} \right){d_1}}}{{2{a_2} + \left( {n - 1} \right){d_2}}} = \frac{{3n + 5}}{{5n + 7}}$$
$${\text{Substituting n = 2n - 1, then}}$$
$$\frac{{2{a_1} + \left( {2n - 2} \right){d_1}}}{{2{a_2} + \left( {2n - 2} \right){d_2}}} = $$ $$\frac{{3\left( {2n - 1} \right) + 5}}{{5\left( {2n - 1} \right) + 7}}$$
$$ \Rightarrow \frac{{{a_1} + \left( {n - 1} \right){d_1}}}{{{a_2} + \left( {n - 1} \right){d_2}}} = $$ $$\frac{{6n - 3 + 5}}{{10n - 5 + 7}}$$ (Dividing by 2)
$$\eqalign{ & \Rightarrow \frac{{{a_{1n}}}}{{{a_{2n}}}} = \frac{{6n + 2}}{{10n + 2}} \cr & \Rightarrow \frac{{{a_{1n}}}}{{{a_{2n}}}} = \frac{{3n + 1}}{{5n + 1}} \cr} $$
This Question Belongs to Arithmetic Ability >> Progressions
The Problem
You are given the ratio of the sums of n terms of two different A.P.s:
Ratio of Sums = (3n + 5) / (5n + 7)
You need to find the ratio of their nth terms.
The Shortcut
To find the ratio of the nth terms from the ratio of the sums, you simply replace n with (2n - 1) in the given expression.
Step-by-Step Solution
Start with the given ratio of sums:
(3n + 5) / (5n + 7)
Apply the substitution n = 2n - 1:
Ratio of nth terms = [ 3(2n - 1) + 5 ] / [ 5(2n - 1) + 7 ]
Simplify the numerator (the top part):
3(2n - 1) + 5
= 6n - 3 + 5
= 6n + 2
Simplify the denominator (the bottom part):
5(2n - 1) + 7
= 10n - 5 + 7
= 10n + 2
Form the final ratio and simplify it:
Ratio = (6n + 2) / (10n + 2)
You can factor out a '2' from both the top and bottom:
Ratio = [ 2(3n + 1) ] / [ 2(5n + 1) ]
Cancel out the '2's:
Ratio = (3n + 1) / (5n + 1)
Therefore, the correct option is B.