Prove by mathematical induction that 7^(2n) - 48n - 1 is a multiple of 2304
I am ok with the first and second steps, but I am confused with the third i.e. when substituting n = k + 1.
Any help please?
Thanks a lot
hi yobacul,
F(n)
$\displaystyle 7^{2n}-48n-1$
is a multiple 2304 ?
F(k+1)
$\displaystyle 7^{2(k+1)}-48(k+1)-1$
is a multiple of 2304 if the "k"th term is ?
Does the hypothesis F(n) cause this to be true ?
$\displaystyle 7^{2k+2}-48k-48-1=7^2\left(7^{2k}\right)-48k-49$
$\displaystyle =7^2\left(7^{2k}\right)-48k-7^2=7^2\left(7^{2k}-1\right)-48k$
Now express -48k as a multiple of $\displaystyle 7^2$
To do this we need to subtract another 48(48k)
{and therefore also add that amount}
$\displaystyle 7^2\left(7^{2k}-1\right)-(48k)-48(48k)+48(48k)$
$\displaystyle =7^2\left(7^{2k}-1\right)-49(48k)+48(48k)$
$\displaystyle =7^2\left(7^{2k}-48k-1\right)+48^2k$
The final term is $\displaystyle 48^2k=2304k$
therefore, the term-by-term link is established.
Hence, if F(n) is valid, F(n+1) also is