-. Use mathematical induction to prove each of the following:* (a) For each natural number n, 2+5+8+...+(3n − 1) =n (3n + 1)2(b) For each natural number n, 1 +5+9+ ·+(4n − 3) = n (2n-1).(c) For each natural number n, 13³ +2³+3³ +...+n³[n(n+1)· ["("+D]².2=

(a). To prove that for each natural number n, 2+5+8+. . . +(3n-1)=n(3n+1)2.

Answered: . Use mathematical induction to prove…

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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2. Use mathematical induction to prove that the statements are true for every positive integer n.(a) 1 + 3 + 6 + . . . +? (?+1)2=? (?+1)(?+2)6(b) 5 + 10 + 15 + . . . + 5? =5? (?+1)2(c) 12 + 23+ . . . + ?3 =?2(?+1)24(d) 14 + 24+ . . . + ?4 =?(?+1)(2?+1) (3?2 +3?−1)30(e) 1 + ? + ?2 + . . . + ??−1 =??−1?−1??? ? ≠ 0, ? ≠ 1(f) 32? + 7 is divisible by 8(g) 13? + 6?is divisible by 7(h) 25?+1 + 52?+2is divisible by 27(i) 72? + 16? − 1 is divisible by 64(j) ?3 − ? is divisible by 3
(c) Prove that n? + 1 is not divisible by 3 for every integer n.
b. Use the Mathematical Induction to show that the given statement is true for all n ≥Z+ P(n): 3+5+7++ (2n + 1) = n(n + 2)
Use mathematical induction to prove the statement for all natural numbers n: 1+4+7+10+...+(3n-2)=n/2(3n-1)
Discrete math.
If n is an odd integer, then the floor of n/2 = (n-1)/2.
You are asked to prove that 12+ 22 + 32 + ·..+n? n(n+1)(2n+1) %3D using the Principle of Mathematical Induction. Which of the following is the correct assumption for the inductive step? O Assume that the equation (k-1)[(k-1)+1][2(k-1)+1] 1 + 22 + 32 + ·..+ (k – 1)² = (k-1)(k)(2k-1) %3D 6 is true. O Assume that the equation 12 = 1 = 123 1(1+1)(2-1+1) %3D is true. O Assume that the equation 12 + 22 + 32 + ..+ k² + (k + 1)² [k+1][(k+1)+1][2(k+1)+1] (k+1)(k+2)(2k+3) is true. O Assume that the equation 12 + 22 + 32 + ...+ k² = k(k+1)(2k+1) %3D 6. is true.
a) Prove by induction that 2 · 2 + 3· 22 + 4· 23 + .. (n + 1) · 2" = n· 27+1 for all n 2 2.
Give an example of (a) a composite number of the form 2º – 1, where p is an odd prime. (b) a composite number of the form 22" + 1, where n is a nonnegative integer.
Prove, using mathematical induction that for an integer n 2 1, 1+5+9+... + (4n – 3) = n (2n – 1).

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. Use mathematical induction to prove each of the following: * (a) For each natural number n, 2+5+8+...+(3n − 1) = = (b) For each natural number n, 1 +5+9+ + (4n − 3) = n (2n − 1). n (3n + 1) 2 n (n+1)]² 2 (c) For each natural number n, 1³ +2³ +33 +...+n n³ = .