Question

In and AP. 5 times of the fifth term is equal to 7 times of the seventh term. prove that 12 term is 0

Answer 1

Given :  5 a₅ = 7 a₇

To Prove : a₁₂ = 0

Proof :  

              Let  a₁ , a₂ , a₃ ....... aₙ are the 1st, 2nd , 3rd ..... nth terms of AP.

                      d = common difference

                      n = natural no.

                  5.a₅ = 7.a₇

              ⇒ 5 { a₁ + ( 5-1)d } = 7 { a₁ + ( 7-1)d}

              ⇒ 5a₁ + 20d = 7a₁ + 42d

              ⇒  2a₁ + 22d = 0

              ⇒ a₁ + 11d = 0                _______________(1)

Now ,

           a₁₂ = a₁ + ( 12 - 1) d

                 = a₁ + 11d

                 = 0        { from above equation 1 }

HENCE PROVED.

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