Answer:
We have to use the mathematical induction to prove the statement is true for all positive integers n.
The integer
is divisible by 3 for every positive integer n.
is divisible by 3.
Hence, the statement holds true for n=1.
- Let us assume that the statement holds true for n=k.
i.e.
is divisible by 3.---------(2)
- Now we will prove that the statement is true for n=k+1.
i.e.
is divisible by 3.
We know that:
![(k+1)^3=k^3+1+3k^2+3k](https://tex.z-dn.net/?f=%28k%2B1%29%5E3%3Dk%5E3%2B1%2B3k%5E2%2B3k)
and ![2(k+1)=2k+2](https://tex.z-dn.net/?f=2%28k%2B1%29%3D2k%2B2)
Hence,
![(k+1)^3+2(k+1)=k^3+1+3k^2+3k+2k+2\\\\(k+1)^3+2(k+1)=(k^3+2k)+3k^2+3k+3=(k^3+2k)+3(k^2+k+1)](https://tex.z-dn.net/?f=%28k%2B1%29%5E3%2B2%28k%2B1%29%3Dk%5E3%2B1%2B3k%5E2%2B3k%2B2k%2B2%5C%5C%5C%5C%28k%2B1%29%5E3%2B2%28k%2B1%29%3D%28k%5E3%2B2k%29%2B3k%5E2%2B3k%2B3%3D%28k%5E3%2B2k%29%2B3%28k%5E2%2Bk%2B1%29)
As we know that:
was divisible as by using the second statement.
Also:
is divisible by 3.
Hence, the addition:
is divisible by 3.
Hence, the statement holds true for n=k+1.
Hence by the mathematical induction it is proved that:
The integer
is divisible by 3 for every positive integer n.
Answer:
62
Step-by-step explanation:
it has to add up to 180 do 118 plus 62 is 180