Question

1^+2^+3^+ ------ +n(n+1)(2n+1)
----------------------
6​

Answer 1

Step-by-step explanation:

. 1

2 + 22 + 32 + · · · + n

2 =

n(n + 1)(2n + 1)

6

Proof:

For n = 1, the statement reduces to 12 =

1 · 2 · 3

6

and is obviously true.

Assuming the statement is true for n = k:

1

2 + 22 + 32 + · · · + k

2 =

k(k + 1)(2k + 1)

6

, (1)

we will prove that the statement must be true for n = k + 1:

1

2 + 22 + 32 + · · · + (k + 1)2 =

(k + 1)(k + 2)(2k + 3)

6

. (2)

The left-hand side of (2) can be written as

1

2 + 22 + 32 + · · · + k

2 + (k + 1)2

.

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