Home
Class 12
MATHS
Prove the following by using iduction fo...

Prove the following by using iduction for all `n in N`. `1+2+3+.....+n=(n(n+1))/(2)`

Text Solution

Verified by Experts

Let `P(n):1+2+3+....+n(n(n+1))/(2)=1`
Step I For `n=1`,
LHS of Eq. (i) =1
RHS of Eq. 9i) `=(1(1+1))/(2)=1`
LHS = RHS
Therefore , P(1) is true.
Step II Let us assume that the result is true for `n=k` , Then ,
P(k) `: 1 + 2 + 3 + ... + k = (k(k+ 1))/(2)`
Step III For n = k + 1 , we have to prove that
P(k+ 1) = ` 1 + 2 + 3 + ... + k + (k + 1) = ((k + 1) (k + 2))/(2)`
L.H.S = 1 + 2 + 3 + ... + k+ ( k + 1)
` = (k(k + 1))/(2) + k + 1 ` [ By assumption step]
`= (k + 1) ((k)/(2) + 1) = (k + 1) ((k+ 2)/(2))`
` = (( k + 1 ) (k + 2))/(2)`
= RHS
This show that the result is true for n = k + 1 . Therefore , by the principle of mathematical induction , the result is true for all `n in N`
Promotional Banner

Similar Questions

Explore conceptually related problems

Prove the following by using the principle of mathematical induction for all n in N 1.2.3.+2.3.4+….+n(n+1)(n+2) = (n(n+1)(n+2)(n+3) )/(4)

Prove the following by using the principle of mathematical induction for all n in N 1+3+3^2 +…………+3^(n-1) = (3^n -1)/(2)

Prove the following by using the principle of mathematical induction for all n in N 1^2 +3 ^2 + 5^2 +………..+ (2n-1)^2 = (n(2n - 1)(2n+1))/(3)

Prove the following by using the principle of mathematical induction for all n in N 1.2+2.3+3.4+…….+n.(n+1) = [ (n(n+1)(n+2))/(3)]

Prove the following by using the principle of mathematical induction for all n in N 1.3+2.3^2+3.3^3 + ……….+n.3^n = ((2n-1)3^(n+1) + 3)/(4)

Prove the following by using the principle of mathematical induction for all n in N 1.2+2.2^2 + 3.2^3 + ……….+ n.2^n =(n-1) 2^(n+1) + 2

Prove the following by using the principle of mathematical induction for all n in N 1+2+3+…….+n lt 1/8 (2n+1)^2

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

Prove the following by using the principle of mathematical induction for all n in N 1 + 2 + 2^2 + …..+2^n = 2^(n+1) -1

Prove the following by using the principle of mathematical induction for all n in N (2n+1) lt 2^n , n >= 3