Prove this by Mathematical Induction?
2*7^n+3*5^n-5 is divisible by 24 ninNN
2*7^n+3*5^n-5 is divisible by 24ninNN
1 Answer
See explanation...
Explanation:
Let
2*7^n+3*5^n-5 is divisible by24
Base case
Putting
2*7^color(blue)(0)+3*5^color(blue)(0)-5 = 2+3-5 = 0
which is divisible by
So
Induction step
Suppose
That is:
2*7^n+3*5^n-5 = 24k
for some integer
Then:
2*7^(n+1)+3*5^(n+1)-5
= 7 * (2*7^n) + 5 * (3 * 5^n) - 5
= 7 * (2*7^n) + 7 * (3 * 5^n) - 7*5 + 35 - 2 * (3 * 5^n) - 5
= 7 * (2*7^n + 3*5^n-5) + 30 - 2 * (3 * 5^n)
= 7 * 24k - 2*(3*5^n-15)
= 7 * 24k - 2*3*5(5^(n-1)-1)
Note that:
(x+1)^n = sum_(k=0)^n ((n),(k)) x^(n-k) = x sum_(k=0)^(n-1) ((n),(k)) x^(n-k) + 1
Hence:
5^(n-1)-1 = (4+1)^(n-1)-1 = 4m+1-1 = 4m
for some integer
So:
2 * 3 * 5*(5^(n-1)-1) = 2 * 3 * 5 * 4m = 24(5m)
for some integer
Thus:
Conclusion
Having shown