Question

Prove each problem, prove by induction

3) Statementn-1 5 25(2m-1) forn2 1 4 Statement Suppose: bo1 . b,-2b-1 + 1 for t 1 en fort >

Answer 1

Let p_n be the statement that n - 1 Sigma k = 0 5.2^k = 5 (2^n - 1) for n greaterthanorequalto 1 Now, for n = 1 1 - Sigma k = 0 5.2^k = 5.2^0 = 5 Also, 5 (2^n - 1) = 5 (2^1 - 1) = 5 (2 - 1) = 5 Therefore the statement is true for n = 1, now, for n = 2 Sigma^1_k = 0 5.2^k = 5.2^0 + 5.2^1 = 5.1 + 5.2 = 5 + 10 = 15 Also, 5 (2^n - 1) = 5.(2^2 - 1) = 5.3 = 15 Therefore the statement is the for n = 2. Let us assume that the result is the for an positive integer m (< n) Therefore m - 1 Sigma k = 0 5.2^k = 5.(26m - 1) Now, for n = m + 1 m Sigma k = 0 5.2^k = m - 1