Home
Class 12
MATHS
Prove 1.4.7+2.5.8+3.6.9+....... upto n t...

Prove `1.4.7+2.5.8+3.6.9+.......` upto n terms `=(n)/(4)(n+1)(n+6)(n+7)`

Text Solution

Verified by Experts

Let `P(n):1.4.7+2.5.8+3.6.9+......+` upto n terms
`(n)/(4)(n+1)(n+6)(n+7)`
i.e., `P(n):1.4.7+2.5.8+3.6.9+......+n(n+3)(n+6)=(n)/(4)(n+1)(n+6)(n+7)`
Step I For `n=1`,
LHS of Eq. (i) `=1.4.7=28`
RHS of Eq. (i) `=(1)/(4)(1+1)(1+6)(1+7)=(2.7.8)/(4)=28`
LHS = RHS
Therefore, P(1) is true .
Step II Let us assume that the result is true for `n=k`. Then , `P(k):1.4.7+2.5.8+3.6.9+......+k(k+3)(k+6)=(k)/(4)(k+1)(k+6)(k+7)`
Step III For `n=k+1`, we have to prove that
`P(k+1):1.4.7+2.5.8+3.6.9+.....+k(k+3)(K+6)+(k+1)(k+4)(k+7)`
`=((k+1))/(4)(k+2)(k+7)(k+8)`
LHS =1.4gt7+2.5.8+3.6.9+......+k(k+3)(k+6)+(k+1)(k+4)(k+7)`
=(k)/(4)(k+1)(k+6)(k+7)+(k+1)(k+4)(k+7)`[by assumption step ]
`=(k+1)(k+7){(k)/(4)(k+6)+(k+4)}`
`=(k+1)(k+7){(k^2+6k+4k+16)/(4)}`
`=(k+1)(k+7){(k^2+10k+16)/(4)}`
`=(k+1)(k+7){((k+2)(k+8))/(4)}`
`=((k+1))/(4)(k+2)(k+7)(k+8)=RHS`
This shows that the result is true for `n=k+1`. Hence, by the principle of mathematical induction , the result is true for all `n in N`.
Promotional Banner

Similar Questions

Explore conceptually related problems

1+3+7+15+31+...+ to n terms

3 + 6+ 9 + …….Find sum of n terms.

Find the sum of series upto n terms ((2n+1)/(2n-1))+3((2n+1)/(2n-1))^2+5((2n+1)/(2n-1))^3+...

Using the principle of mathematical induction, prove that : 1. 2. 3+2. 3. 4++n(n+1)(n+2)=(n(n+1)(n+2)(n+3))/4^ for all n in N .

P(n) : 1^(2) + 2^(2) + 3^(2) + .......+ n^(2) = n/6(n+1) (2n+1) n in N is true then 1^(2) +2^(2) +3^(2) + ........ + 10^(2) = .......

Prove that 1^2/(1.3)+2^2/(3.5)+3^2/(5.7)+.....+n^2/((2n-1)(2n+1))=((n)(n+1))/((2(2n+1))

prove that 7 + 77 + 777 +...... + 777........._(n-digits) 7 = 7/81 (10^(n+1) - 9n - 10) for all n in N

Prove that 1.3+3.5+5.7+......+(2n-1)(2n+1)=(n(4n^2+6n-1))/3

Find the sum of n terms of the series 1^3+3.2^2+3^3+3.4^2+5^3+3.6^2+....... when (i)n is odd (ii)n is even

Prove the following by using the principle of mathematical induction for all n in N (1)/(2.5) + (1)/(5.8) + (1)/(8.11) + ……. + (1)/((3n-1)(3n+2)) =(n)/((6n + 4))