Mathamatical Inductions Finding right part of the equality?

Question

heres an example question

Find the right part of the equality.

2^3 + 4^3 + 6^3. . .(2n)^3 = _______


I know the answer is 2n^2(n+1)^2


The problem is that i dont know how to get there. My teacher only did the proveing part and now this was on the assignment.

Just so you guys know, 2n^2(n+1)^2 was not in the question.so you cant use that to solve the problem .. .

as i said before, 2n^2(n+1)^2 is NOT in the question there fore CANNOT be used in the equation. 2n^2(n+1)^2 is the ANSWER.

Answer 1

2³ + 4³ + 6³ + ... (2n)³ = ∑ (2i) with limits 1 and n which I will write as ∑ (2i) [1,n]

suppose for some n, n = k

then ∑ (2i) [1,k] = 2k² (k + 1)² is assumed to be true

adding next term to both sides

∑ (2i) [1,k] + (2(k + 1))² = 2k² (k + 1)² + (2(k + 1))²

.......................................... = 2(k + 1)² [ k² + 4(k + 1)]

.......................................... = 2(k + 1)²(k + 2)²

.......................................... = ∑ (2i) [1,(k+ 1)]

Therefore k is true implies (k+1) is true .......................A

when k = 1

∑ (2i) [1,k] = 2³ = 8

2k² (k + 1)² = 2 . 1 . 4 = 8

Therefore 1 is true

From line A ... 1 is true implies 2 is true .....implies n is true

Therefore 2³ + 4³ + 6³ + ... (2n)³ = 2n² (n + 1)²

Answer 2

Could this question be re-phrased to read:-
Prove by induction that:-
P(n): 2³ + 4³ + 6³ -------(2n)³ = 2n².(n + 1)²
If so the method is as follows:-
Assume that P(n) is true for n = k
ie assume P(k) is true where:-
P(k): 2³ + 4³ + 6³ ---(2k)³ = 2k².(k + 1)²
Have now to show that P(1) and P(k + 1) are true.
Consider P(1)
LHS = 2³ = 8
RHS = 2.(1 + 1)² = 2 x 4 = 8
Thus P(1) is true.
Consider P(k + 1):-
2³ + 4³ + 6³ + ---2.(k + 1)³= 2(k+ 1)².(k + 2)²
Have now to prove this:-
Now
2³ + 4³ + 6³ + --(2k)³ = 2k².(k + 1)² is true
2³ + 4³ + --(2k)³ + (2k + 2)³ = 2k².(k + 1)² + (2k + 2)³
= 2k² .(k + 1)² + 2³.(k + 1)³
= 2.(k + 1)².(k² + 4k + 4)
= 2.(k + 1)² .(k + 2)²
Thus P(k + 1) is true.
Now have:-
P(k) true
P(1) true
P(k + 1) true
THEREFORE P(n) IS TRUE.