[Math Lair] Pillow-Problems: Problem #32

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Problem #31 | Problem #38
For more information on this collection, see Pillow-Problems by Charles L. Dodgson (Lewis Carroll).

Problem:

32.

Sun the Series 1 · 5 + 2 · 6 + &c. (1) to n terms; (2) to 100 terms.

[7/4/89

Answer:

32.

(1)
n · (n + 1) · (2n + 13)
6
;

(2) 358550.

Solution:

32.

The nth term is n · (n + 4);

∴ the (n + 1)th term is (n + 1) · (n + 5) = (n + 1) · {(n + 2) + 3},

= (n + 1) · (n + 2) + 3(n + 1);

Sn =
n · (n + 1) · (n + 2)
3
+ 3 ·
n · (n + 1)
2
+ C; and C = 0;

Sn = n · (n + 1) · (
n + 2
3
+
3
2
) =
n · (n + 1) · (2n + 13)
6
.

Q.E.F.