Question

prove by mathematical induction. 5+10+15+.....+5n=5n(n+1)/2.​

Answer 1

Step-by-step explanation:

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Answer 2

The prove of the given statement by mathematical induction is shown in explanation part.

Step-by-step explanation:

The given statement is

$5+10+15+...+5n=\frac{5n(n+1)}{2}$

For n = 1

$5=\frac{5\cdot1(1+1)}{2}\\\\5=5\cdot1\\\\5=5=>\text{ True}$

Hence, the given statement is true for n = 1

Let the given statement is true for n = k

So, we have

$P(k)=5+10+15+...+5k-5+5k=\frac{5k(k+1)}{2}$

Now, we have to prove that the statement is true for n = k+1

Substitute n = k+1

$5+10+15+...+\frac{5(k+1)(k+1+1)}{2}\\\\5+10+15+...+\frac{(5k+5)(k+2)}{2}\\\\5+10+15+...+\frac{5k^2+15k+10}{2}\\\\5+10+15+...+\frac{5k(k+3)}{2}+5\\\\5+10+15+...+\frac{5k(k+1}{2}+\frac{10k}{2}+5\\\\5+10+15+...+\frac{5k(k+1}{2}+5k+5\\\\P(k)+5(k+1)$

It means that it is true for n = k+1 as well

Hence, from mathematical induction, we can say that

$5+10+15+...+5n=\frac{5n(n+1)}{2}$

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