Correct Answer: A
Explanation:
n = 1
S(1) : 1=1(2(1) -1) Þ 1=1
Thus condition I is satisfied
Now suppose that S(n) is true for n = k
S(k) : 1+ 5 + 9 + ..............+ (4k - 3) = k (2k -1) …………. (i)
The statement for n = k +1 becomes
S(k +1) : 1+ 5 + 9 + ..............+ (4(k +1) - 3) = (k +1)(2(k +1) -1)
Þ 1+ 5 + 9 + ..............+ (4k +1) = (k +1)(2k + 2 -1)
= (k +1)(2k +1)
= 2k 2 + 2k + k +1
= 2k 2 + 3k +1
Adding 4k +1 on both sides of equation (i)
1+ 5 + 9 + ..............+ (4k - 3) + (4k +1) = k (2k -1) + 4k +1
Þ 1+ 5 + 9 +..............+ (4k +1) = 2k 2 - k + 4k +1
= 2k 2 + 3k +1
Thus S(k +1) is true if S(k) is true, so condition II is satisfied and S(n) is true
for all positive integer n.