prove that 1^2+3^2+5^2+......+(2n-1)^2n(2n-1)(2n+1)/3
Answers
Answer:
Step-by-step explanation:
1
2
+3
2
+5
2
...+(2n−1)
2
=
3
n(2n−1)(2n+1)
∀n∈N
PROOF:
P(n)=1
2
+3
2
+5
2
...+(2n−1)
2
=
3
n(2n−1)(2n+1)
P(1):(2×1−1)
2
=
3
1(2−1)(2+1)
⇒(1)
2
=1=
3
1×1×3
=1
∴ L.H.S=R.H.S (Proved)
∴P(1) is true.
Now, let P(m) is true.
Then, P(m)=1
2
+3
2
+5
2
...+(2m−1)
2
=
3
m(2m−1)(2m+1)
Now, we have to prove that P(m+1) is also true.
P(m+1)=1
2
+3
2
+5
2
...+(2m−1)
2
+[2(m+1)−1]
2
=P(m)+(2m+2−1)
2
=P(m)+(2m+1)
2
=
3
m(2m−1)(2m+1)
+(2m+1)
2
=
3
m(2m−1)(2m+1)+3(2m+1)
2
=
3
(2m+1)[m(2m−1)+3(2m+1)]
=
3
(2m+1)[2m
2
−m+6m+3]
=
3
(2m+1)[2m
2
+5m+3]
=
3
(2m+1)[2m
2
+2m+3m+3]
=
3
(2m+1)[2m(m+1)+3(m+1)]
=
3
(2m+1)(2m+3)(m+1)
=
3
(m+1)[2(m+1)+1][2(m+1)−1]
∴p(m+1) is also true (Proved) pls mark me as brainliest