Explanation
Solution
The formula for the sum of an infinite Geometric Progression (GP) is S∞=1−ra, where |r| < 1.
1. Evaluation of (A):
-
First term a=2+1
-
Common ratio r=2+11=(2+1)(2−1)2−1=2−1
-
S∞=1−(2−1)2+1=2−22+1
-
Rationalize: 2(2−1)2+1×2+12+1=2(2−1)(2+1)2=22+1+22=23+22=232+4
Match: (A) → IV
2. Evaluation of (B):
This is a sum of two infinite GPs:
Series 1: 21+231+251⋯⟹a=21,r=41⟹S1=1−1/41/2=3/41/2=32
Series 2: 321+341+361⋯⟹a=91,r=91⟹S2=1−1/91/9=8/91/9=81
3. Evaluation of (C):
4. Evaluation of (D):
Final Matching:
(A)-IV, (B)-I, (C)-II, (D)-III
Correct Option: (b)
