If S1 is the sum of an arithmetic progression of ‘n’ odd number of terms and S2 is the sum of the terms of the series in odd places, then $$\frac{{{S_1}}}{{{S_2}}}$$
A. $$\frac{{2n}}{{n + 1}}$$
B. $$\frac{n}{{n + 1}}$$
C. $$\frac{{n + 1}}{{2n}}$$
D. $$\frac{{n - 1}}{n}$$
Answer: Option A
Solution(By Examveda Team)
Odd numbers are 1, 3, 5, 7, 9, 11, 13, ...... n∴ S1 = Sum of odd numbers = n2
S2 = Sum of number at odd places
3, 7, 11, 15, ......
a = 3, d = 7 - 3 = 4 and number of term = $$\frac{n}{2}$$
$$\eqalign{ & {S_2} = \frac{n}{{2 \times 2}}\left[ {2 \times 3 + \left( {\frac{n}{2} - 1} \right) \times 4} \right] \cr & \,\,\,\,\,\,\,\,\, = \frac{n}{4}\left[ {6 + 2n - 4} \right] \cr & \,\,\,\,\,\,\,\,\, = \frac{n}{4}\left[ {2n + 2} \right] \cr & \,\,\,\,\,\,\,\,\, = \frac{{n\left( {n + 1} \right)}}{2} \cr & \therefore \frac{{{s_1}}}{{{s_2}}} = \frac{{{n^2} \times 2}}{{n\left( {n + 1} \right)}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{{2n}}{{n + 1}} \cr} $$
This Question Belongs to Arithmetic Ability >> Progressions
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Here, what do indicate odd places? Aren't the term numbers?
1 stays at 1st place,
5 at 3rd,
9 at 5th ....and so on.
aren't the odd term number indicating the odd places? So, why 3,7,11,15... are taken as odd place holders?