Prove by induction that for all positive integers n , f(n) = 2^(3n+1) + 3*5^(2n+1) , is divisible by 17.

  1. Prove the base caseFor n=0, f(0)= 2 + 15 = 17Therefore, when n=0, f(n) is divisible by 17, base case is true2. Assume true for any integerAssume for n=k, f(k) is divisible by 17f(k)= 23k+1 + 3(52k+1) ;3. Work out function for the next integerf(k+1) = 23k+4 + 3(52k+3) = 8(23k+1 + 3(52k+1)) + 25552k255 = 1517, therefore the second term is divisible by 1723k+1 + 3(52k+1) = f(k), so if f(k) is divisible by 17, f(k+1) is divisible by 17.Since f(0) is divisible by 17, and if f(k) is divisible by 17, then f(k+1) is divisible by 17, then f(n) is divisible by 17 for all positive integers.

Related Further Mathematics A Level answers

All answers ▸

Find the area of the surface generated when the curve with equation y=cosh(x) is rotated through 2 pi radians about the x axis, with 2<=x<=6


Compute the derivative of arcsin(x).


What IS a Taylor Series?


How to multiply and divide by complex numbers


We're here to help

contact us iconContact usWhatsapp logoMessage us on Whatsapptelephone icon+44 (0) 203 773 6020
Facebook logoInstagram logoLinkedIn logo

© MyTutorWeb Ltd 2013–2025

Terms & Conditions|Privacy Policy
Cookie Preferences