Question

sigma I=1 to n i*2^i=2+(n-1)*2^n+1 prove by mathematical induction
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Answer 1

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11th

Maths

Principle of Mathematical Induction

Introduction to Mathematical Induction

Prove by mathematical induc...

MATHS

Prove by mathematical induction that

n. 1 + ( n - 1) 2 + (n - 2) 3 + ..... + 2 (n -1) + 1.n =

6

n

(n + 1)(n + 2)

MEDIUM

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ANSWER

Result is obviously true for n=1 Assume P(n) i.e.

P(n)=n⋅1+(n−1)⋅2+....+2(n−1)++1⋅n

=

6

n

(n+1)(n+2)

p(n + 1) = (n + 1) 1 + n 2+ ,... 3(n -1)+2n + 1(n +1)

∴p(n+1)−p(n)=1⋅1+1⋅2+1⋅3+1(n−1)+1⋅n+1+1⋅(n+1)

= 1 + 2 + 3 ... + n + 1(n + 1)

=

2

n(n+1)

+n+1=

2

(n+1)(n+2)

∴p(n+1)=

2

(n+1)(n+2)

p(n)

2

(n+1)(n+2)

+

6

(n+1)(n+2)

2

(n+1)(n+2)

1+[

3

n

]

6

(n+1)(n+2)(n+3)

=

6

n

(n+1)(n+2)