By the principle of mathematical induction, prove that 10n + 3*4n+2 + 5 is divisible by 9 - Maths - Principle of Mathematical Induction

Question

By the principle of mathematical induction, prove that
10n + 3*4n+2 + 5 is divisible by 9

Uday Kiran · Accepted Answer

$n n L n e n t n â¢ n P n (n n n) n = n n^{n 10 n n n n n n} n + n 3 n \cdot n n^{n 4 n n n n n + n 2 n n} n + n 5 n n n P n u n t n â¢ n n n n n n n n = n 1 n, n n P n (n 1 n) n = n 10 n + n 3 n \cdot n 16 n + n 5 n = n 63 n â¢ n n n n n n w n h n i n c n h n â¢ n n n n i n s n â¢ n n n n d n i n v n i n s n i n b n l n e n â¢ n n n n b n y n â¢ n n n n n â¢ n 9 n n S n o n, n n P n (n 1 n) n n n i n s n â¢ n n n n t n r n u n e n n n n L n e n t n â¢ n n P n (n n n) n = n t n r n u n e n n n s n o n n, n n n^{n 10 n n n n n n} n + n 3 n \cdot n n^{n 4 n n n n n + n 2 n n} n + n 5 n = n 9 n λ n n n N n o n w n, n n l n e n t n s n â¢ n n n n n p n r n o n v n e n â¢ n n n n n f n o n r n â¢ n n n n n P n (n n n + n 1 n) n n P n (n n n + n 1 n) n = n n^{n 10 n n n n n + n 1 n n} n + n 3 n \cdot n n^{n 4 n n n n n + n 1 n + n 2 n n} n + n 5 n n = n 10 n \cdot n n^{n 10 n n n n n n} n + n 3 n \cdot n n^{n 4 n n n n n n} n \cdot n n^{n 4 n n n 3 n n} n + n 5 n n = n 10 n \cdot n n^{n 10 n n n n n n} n + n 192 n \cdot n n^{n 4 n n n n n n} n + n 5 n n = n (n 9 n + n 1 n) n \cdot n n^{n 10 n n n n n n} n + n (n 48 n + n 144 n) n \cdot n n^{n 4 n n n n n n} n + n 5 n n = n 9 n \cdot n n^{n 10 n n n n n n} n + n n^{n 10 n n n n n n} n + n 48 n \cdot n n^{n 4 n n n n n n} n + n 144 n \cdot n n^{n 4 n n n n n n} n + n 5 n n = n 9 n \cdot n n^{n 10 n n n n n n} n + n 144 n \cdot n n^{n 4 n n n n n n} n + n n^{n 10 n n n n n n} n + n 48 n \cdot n n^{n 4 n n n n n n} n + n 5 n n = n 9 n \cdot n n^{n 10 n n n n n n} n + n 144 n \cdot n n^{n 4 n n n n n n} n + n 9 n λ n n = n 9 n (n n^{n 10 n n n n n n} n + n 16 n \cdot n n^{n 4 n n n n n n} n + n λ n) n n n n n n n w n h n i n c n h n â¢ n n n n â¢ n i n s n â¢ n n n n n d n i n v n i n s n i n b n l n e n â¢ n n n n b n y n n â¢ n n n n 9 n n n n n H n e n n n c n e n n, n n P n (n n n + n 1 n) n n n i n s n â¢ n n n n t n r n u n e n â¢ n n n â¢ n f n o n r n â¢ n n n n a n l n l n â¢ n n n n â¢ n n n n â¢ n n n n n n$ hope it helps u