Solveeit Logo
Login

Question

Mathematics Question on Binomial Theorem for Positive Integral Indices

Show that 9n+1−8n−99^{ n+1} -8n -9 is divisible by 6464, whenever n is a positive integer.

Answer

In order to show that 9n+1−8n−99^{n+1}-8n-9 is divisible by 6464, it has to be proved that,
9n+1−8n−99^{n+1}-8n-9 =64k64k, where k is some natural number
By Binomial Theorem,
(1+a)m(1+a)^m = mC0+mC1a+mC2a2+...+mCmam^mC_0+^mC_1a+ ^mC_2 a^2+...+ ^mC_m a^m
For a=8a = 8 and m=n+1m = n+ 1, we obtain
(1+8)n+1(1+8)^{n+1} = n+1C0+n+1C1(8)+n+1C2(8)2+...+n+1Cn+1(8n+1)^{n+1}C_0+ ^{n+1}C_1(8) + ^{n+1}C_2 (8)^2 + ... + ^{n+1}C_{n+1}(8n+1)
⇒ 9n+1=1+(n+1)(8)+82[n+1C2+n+1C3×8+...+n+1Cn+1(8)n−1] 9^{n+1} = 1 + (n+1)(8) + 8^2 [^{n+1}C_2 + ^{n+1}C_3×8+...+ ^{n+1}C_{n+1} (8)^{n-1}]
⇒ 9n+1=9+8n+64[n+1C2+n+1C3×8+...+n+1Cn+1(8)n−1] 9^{n+1} = 9+8n +64[ ^{n+1}C_2 + ^{n+1}C_3 × 8+...+^{n+1}C_{n+1} (8)^{n-1}]
⇒ 9n+1−8n−9=64k9^{n+1}-8n-9=64k, where kk = n+1C2+n+1C3×8+...+n+1Cn+1(8)n−1^{n+1}C_2 + ^{n+1}C_3 × 8+...+^{n+1}C_{n+1} (8)^{n-1} is a natural number.

Thus, 9n+1−8n−99^{n+1}-8n-9 is divisible by 6464, whenever nn is a positive integer.