Let s(n)=1+2+···+ n be the sum of the first n natural numbers and let c(n)=1 3 +2 3 + ···+ n3 be the sum of the first n cubes. prove the following equalitiesby inductionon n, to arriveat thecurious conclusionthatc(n)=s2(n) for every

Step-by-step explanation:

We have

To prove that by induction

Let P(n) be that statement for n

n=1 gives left side =1 =right side

True for n=1

Assume that P(k) is true

i.e.

We have to prove P(k+1) is true assuming P(k)

LHS of

RHS =

since by adding last term with Ck gives the sum

For simplification take (k+1) square outside

RHS =

Thus proved by mathematical induction

start with a line segment pq that we will copy. mark a point r that will be one endpoint of the new line segment. set the compasses' point on the point p of the line segment to be copied. adjust the compasses' width to the point q. the compasses' width is now equal to the length of the line segment pq.

step-by-step explanation: