Question a for n2 prove that 2 2 3 2 4 2 n 2 n1 3 Hint Us

Question: (a) for n>=2, prove that (2 2) + (3 2) + (4 2) +?+(n 2) = (n+1 3) [Hint: Use induction and Pascal?s rule.]

Solution

2C2 + 3C2 + ... nC2 = (n+1)C3

1 + 3 + 6 + 10 + ... n!/(2! * (n-2)!) = (n+1)! / (3! * (n+1-3)!)

1 + 3 + 6 + 10 + .... n(n-1)/2 = (n-1)(n)(n+1)/6

Let us prove the above by induction :

Step 1 : Prove it for n = 2...

2(2-1)/2 = (2-1)(2)(2+1)/6
2(1)/2 = 1*2*3/6
2/2 = 6/6
1 = 1
Proved

Step 2 : Assume it is true for n = k :

1 + 3 + 6 + 10 + .... k(k-1)/2 = (k-1)(k)(k+1)/6

Step 3 : We are to prove that it is true for k+1...

1 + 3 + 6 + 10 + .... k(k-1)/2 + k(k+1)/2 = (k+1-1)(k+1)(k+1+1)/6

(k-1)(k)(k+1)/6 + k(k+1)/2 = k(k+1)(k+2)/6 --> This is to be proved

Take the left and add the fractions :

((k-1)k(k+1) + 3k(k+1)) / 6

((k^2 - k)(k + 1) + 3k^2 + 3k) / 6

(k^3 + 3k^2 + 3k - k) / 6

(k^3 + 3k^2 + 2k) / 6

k(k^2 + 3k + 2) / 6

k(k + 1)(k + 2) / 6

This is the same as the right hand side

So, the relation has been proved!