For any integer n2 1. prove that: (a) 1+2+3+..+ (n – 1) + n + (n – 1) + . .. +3+2+ 1 = n². (b) +...+ n(n + 1) (n + 1)' [Hint: Use the splitting identity 1/k – 1/(k + 1) = 1/k(k + 1) to rewrite the left-hand side.] (c) 1.2+2-3+3 - 4 + · . · + n(n + 1) n(n + 1)(n + 2) 3 [Hint: Use the identity k(k + 1) = k² + k and collect the squares.]

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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11. For any integer n > 1, prove that:
(a)
1+2+3+...+ (n – 1) +n
+ (n – 1) +...+3+2+1 = n?.
+
1.2
2.3
3.4
п(n + 1)
(n + 1)
[Hint: Use the splitting identity
1/k – 1/(k + 1) = 1/K(k + 1)
to rewrite the left-hand side.]
(c) 1.2+2.3+3.4 +...+ n(n + 1)
n(n + 1)(n + 2)
3
[Hint: Use the identity k(k + 1) = k? +k and
collect the squares.]
1-3
1
+
3.5
(d)
1.3
5.7
1
(2n – 1)(2n + 1)
2n +1
Hint: Use the identity
1
(2k – 1)(2k + 1)
2 2k - 1
2k +1
Transcribed Image Text:11. For any integer n > 1, prove that: (a) 1+2+3+...+ (n – 1) +n + (n – 1) +...+3+2+1 = n?. + 1.2 2.3 3.4 п(n + 1) (n + 1) [Hint: Use the splitting identity 1/k – 1/(k + 1) = 1/K(k + 1) to rewrite the left-hand side.] (c) 1.2+2.3+3.4 +...+ n(n + 1) n(n + 1)(n + 2) 3 [Hint: Use the identity k(k + 1) = k? +k and collect the squares.] 1-3 1 + 3.5 (d) 1.3 5.7 1 (2n – 1)(2n + 1) 2n +1 Hint: Use the identity 1 (2k – 1)(2k + 1) 2 2k - 1 2k +1
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