What are the steps to get the answer?

Question

Use Mathematical Induction to prove the sum of Arithmetic Sequences:

Hint: First write down what P(1) says and then prove it. Then write down what P(k+1) says and prove that using the fact that P(k) is true.

P(1) says 1∑j=1(a+(j−1)d)= ?

 P(k+1) says 

Answer 1

I will use L to denote the left-hand side, and will use R to denote

the right-hand side of the identity to be proven.

L(n) = n∑j=1 (a+(j-1)d)

R(n) = (n/2)(2a+(n-1)d)

We are to prove that

L(n) = R(n) for all integers n ≥ 1

BASIS STEP:

L(1) = 1∑j=1 (a+(j-1)d) = (a+(1-1)d) = a

R(1) = (1/2)(2a+(1-1)d) = a

So

L(1) = R(1)

INDUCTION STEP:

Assume L(k) = R(k) for some k and prove L(k+1) = R(k+1)

L(k+1)

= k+1∑j=1 (a+(j-1)d)

= k∑j=1 (a+(j-1)d) + (a+(k+1-1)d)

= L(k) + (a+kd)

= R(k) + (a+kd)

= (k/2)(2a+(k-1)d) + (a+kd)

= (k/2)(2a+kd) - (k/2)d + (a+kd)

= (k/2)(2a+kd) + a+(k/2)d

R(k+1)

= (k+1/2)(2a+(k+1-1)d)

= (k+1/2)(2a+kd)

= (k/2)(2a+kd) + (1/2)(2a+kd)

= (k/2)(2a+kd) + a+(k/2)d

So L(k+1) = R(k+1)

which completes the proof.