Question

श्न25. गणितीय आगमन विधि के प्रयोग द्वारा सिद्ध कीजिये कि
13 +23+23
- -+n3
n(n.+1)
जहाn.EN
2
2​

Answer 1

Given :  1³  + 2³ + 3³  +........................+ n³  = {(n)(n + 1)/2}²

To Find : गणितीय आगमन विधि के प्रयोग द्वारा सिद्ध कीजिये

Prove by Mathematical Induction

Solution:

P(n) = 1³  + 2³ + 3³  +........................+ n³  = {(n)(n + 1)/2}²

n = 1

P(1) = 1³ = 1

{(n)(n + 1)/2}²  = { 1(1 + 1)/2}² = 1

n = k

P(k)  = 1³  + 2³ + 3³  +........................+k³  =  {(k)(k + 1)/2}²

n = k + 1

P(k+1)  = 1³  + 2³ + 3³  +........................+k³+(k+1)³  =  {(k)(k + 1)/2}²  =  {(k+1)(k+1 + 1)/2}²

LHS  = 1³  + 2³ + 3³  +........................+k³+(k+1)³

=  {(k)(k + 1)/2}²  + (k+1)³

= {(k + 1)/2}²{ k²  + 4(k+1)}

=  {(k + 1)/2}²{ k²  + 4k + 4}

=  {(k + 1)/2}²{ (k+2)² }

= {(k + 1)(k+2)/2}²

RHS

=  {(k+1)(k+1 + 1)/2}²

= {(k + 1)(k+2)/2}²

LHS = RHS

QED

Hence Proved

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