Use mathematical induction to prove the following.6 + 12 + 18 + . . . + 6n = 3n(n + 1)
What will be an ideal response?
Answers may vary. One possibility:
Sn: 6 + 12 + 18 + . . . + 6n = 3n(n + 1)
S1: 6 = 3 ? 1 ? (1 + 1)
Sk: 6 + 12 + 18 + . . . + 6k = 3k(k + 1)
Sk+1: 6 + 12 + 18 + . . . + 6k + 6(k + 1) = 3(k + 1)(k + 2)
Step 1: Since 3 ? 1 ? (1 + 1) = 3 ? 2 = 6, S1 is true.
Step 2: Let k be any natural number. Assume Sk. Deduce Sk+1.
6 + 12 + 18 + . . . + 6k = 3k(k + 1) By Sk
6 + 12 + 18 + . . . + 6k + 6(k + 1) = 3k(k + 1) + 6(k + 1) Adding 6(k + 1)
6 + 12 + 18 + . . . + 6k + 6(k + 1) = (3k + 6)(k + 1) Distributive law
6 + 12 + 18 + . . . + 6k + 6(k + 1) = 3(k + 2)(k + 1)
6 + 12 + 18 + . . . + 6k + 6(k + 1) = 3(k + 1)(k + 2).
You might also like to view...
Perform the indicated operations.A current of 2.64 e-1.48j amperes flows in a given circuit. Express this current in polar form.
A. -1.48 (cos 2.64° + j sin 2.64° ) B. -1.48 (cos 151.3° + j sin 151.3° ) C. 2.64 (cos (-1.48°) + j sin (-1.48° )) D. 2.64 (cos (-84.8°) + j sin (-84.8° ))
Use the order of operations to evaluate each problem.
A. 63 B. 21 C. 50 D. 7
Verify the identity.csc x - sin x = cos x cot x
What will be an ideal response?
Simplify the expression. Include absolute value bars where necessary.
A. x + 1
B. 
+ 1
C. ±x + 1
D. 