Question

Let n be a positive integer. Let S(n) = n sigma j=1 ((1/3j − 2) − (1/3j + 1)). a) Compute the value of S(1), S(2), S(3), and S(4). b) Make a conjecture that gives a closed form (i.e., not a summation) formula for the value of S(n). c) Use induction to prove your conjecture is correct.

Answer 1

Given the Series is -

(a) To compute the value of S(1) , S(2) , S(3) annd S(4) , we will just replace n in the above series by 1 , 2 , 3 and 4 , one by one .

So , we have -

For n = 1 :

i.e.,

For n = 2 , we have -

For n = 3 , we have -

Similarly , for n = 4 , we have -

(b) : we have -

In order find the closed form of the above series , we will use the formula -

Closed   

Where ,

n = number of terms

A = first term of the series

L = last term of the series .

Now , we have -

Number of terms in the given series = n

First term of the series , i.e.,

i.e.,

   AND ,

Last term of the series , i.e.,

Thus Closed form is given as-

Hence , the closed form of the above series is

i.e.,

(c) Till now we have , the closed form the given series as -

Now , we will prove that above statement is true using induction .

Let S(n) be a statement given as -

Step 1: Put n = 1 , in above statement -

, which is true .

Thus , S(1) is true , i.e., S(n) is true for n = 1.

Step 2: Let us suppuse that the given statement S(n) is true for some n = k

so that , we have -

.......(1)

Step 3 : We will now show that the given statement is true for n = ( k + 1) also .

For this -

ADD to both sides of the above equation (1) , we get -

Now LHS and RHS of the above equation will transform as -

The LHS of the above equation is clearly the S(k+1) , i.e.,

Simplifying above expression , we get -

Thus , we have -

i.e.,

Hence , it has been proved by induction that if the statement S(n) is true for each n = k , it is true for n = (k+1) also .

Hence , the conjecture of the closed form is correct .