Question

the value is i^ 1+3+5+...+(2n+1)​

Answer 1

Answer:

Sorry I don't see the question anywhere.......

Step-by-step explanation:

Answer 2

Let P(n): 1 + 3 + 5 + ..... + (2n - 1) = n
2
be the given statement
Step 1: Put n = 1
Then, L.H.S = 1
R.H.S = (1)
2
= 1
∴. L.H.S = R.H.S.
⇒ P(n) istrue for n = 1
Step 2: Assume that P(n) istrue for n = k.
∴ 1 + 3 + 5 + ..... + (2k - 1) = k
2

Adding 2k + 1 on both sides, we get
1 + 3 + 5 ..... + (2k - 1) + (2k + 1) = k
2
+ (2k + 1) = (k + 1)
2

∴ 1 + 3 + 5 + ..... + (2k -1) + (2(k + 1) - 1) = (k + 1)
2

⇒ P(n) is true for n = k + 1.
∴ by the principle of mathematical induction P(n) is true for all natural numbers 'n'
Hence, 1 + 3 + 5 + ..... + (2n - 1) =n
2
, for all n ϵ n