Question

if the sum of n terms of an AP is 3n^2 +5n and it's m th term is 164, find the vale of m​

Answer 1

Answer :

m = 27

Step-by-step explanation :

$\bigstar \underline{\underline{\bf Arithmetic \ Progression:}}$

• It is the sequence of numbers such that the difference between any two successive numbers is constant.

• In AP,

     a - first term

     d - common difference

     aₙ - nth term

     Sₙ - sum of n terms

• General form of AP,

        a , a+d , a+2d , a+3d , ..........

• Formulae :-

         nth term of AP,

          $\boxed{\bf a_n=a+(n-1)d}$

         Sum of n terms in AP,

          $\boxed{\bf S_n=\frac{n}{2}[2a+(n-1)d]}$

_________________________

Given,

  Sum of n terms, Sₙ = 3n² + 5n

       

  Sum of (n - 1) terms,

          Sₙ₋₁  = 3(n - 1)² + 5(n - 1)

          Sₙ₋₁  = 3(n² + 1² -2n) + 5(n - 1)

          Sₙ₋₁  = 3(n² + 1 - 2n) + 5(n - 1)

          Sₙ₋₁  = 3n² + 3 - 6n + 5n - 5

          Sₙ₋₁  = 3n² - n - 2

Now,

 nth term = (Sum of n terms) - (Sum of n-1 terms)

          aₙ   = Sₙ - Sₙ₋₁

          aₙ   = 3n² + 5n - (3n² - n - 2)

          aₙ   = 3n² + 5n - 3n² + n + 2

          aₙ   = 6n + 2

∴ nth term, aₙ = 6n + 2

⇒ mth term, aₘ = 6m + 2

Given, mth term = 164

                     aₘ  = 164

              6m + 2 = 164

                    6m = 164 - 2

                    6m = 162

                      m = 162/6

                      m = 27

∴ The value of m is 27