Q. 12
Prove the following by the principle of mathematical induction:
1.3 + 2.4 + 3.5 + … + n . (n + 2) 
Answer :
Let P(n): 1.3 + 2.4 + 3.5 + … + n.(n + 2) = 
For n = 1
P(1): 1.3 = 
.1.(2)(9)
= 3 = 3
Since, P(n) is true for n = 1
Now,
For n = k
= P(n): 1.3 + 2.4 + 3.5 + … + k . (k + 2)= 
- - - - - (1)
We have to show that
= 1.3 + 2.4 + 3.5 + … + k . (k + 2) + (k + 3) = 
Now,
= {1.3 + 2.4 + 3.5 + … + k (k + 2)} + (k + 1)(k + 3)
= 
k(k + 1)(2k + 7) + (k + 1)(k + 3) using equation (1)
= (k + 1)
= (k + 1)
= (k + 1)
= (k + 1)
= (k + 1)
= (k + 1)
= 
(k + 1)(k + 2)(2k + 9)
Therefore, P(n) is true for n = k + 1
Hence, P(n) is true for all n∈ N
Rate this question :
Take the challenge, Quiz on Vectors37 mins
Vectors- Cosine & SIne Rule54 mins
Interactive Quiz on vector addition and multiplications38 mins
Scalar and vector product in one shot56 mins
Revise Complete Vectors & it's application in 50 Minutes51 mins
Scalar and Vector Product49 mins
Addition of Vectors - Kick start Your Preparations47 mins
Quick Revision of Vector Addition and Multiplications37 mins
Quiz | Vector Addition and Multiplications35 mins
Unit Vectors24 mins
Prove that cos α + cos (α + β) + cos (α + 2β) + … + cos (α + (n – 1)β) 
for all n ϵ N
Prove that sin x + sin 3x + … + sin (2n – 1) x 
for all
nϵN.
RD Sharma - MathematicsProve the following by the principle of mathematical induction:
1.2 + 2.3 + 3.4 + … + n(n + 1)
Prove the following by the principle of mathematical induction:
1.3 + 2.4 + 3.5 + … + n . (n + 2)
Prove the following by the principle of mathematical induction:
Prove the following by the principle of mathematical induction:
12 + 32 + 52 + … + (2n – 1)2
Prove the following by the principle of mathematical induction:
a + ar + ar2 + … + arn – 1
Prove the following by the principle of mathematical induction:
2 + 5 + 8 + 11 + … + (3n – 1) = 1/2 n(3n + 1)
RD Sharma - MathematicsProve the following by the principle of mathematical induction:
1.3 + 3.5 + 5.7 + … + (2n – 1) (2n + 1) 
Prove that 
for all natural
numbers, n ≥ 2.
RD Sharma - Mathematics