Question

The 4" term of a G.P. is square of its second term, and the first term 15-3
Determine its 7th term.​

Answer 1

Correct Question :

The 4th term of a G.P. is square of its second term, and the first term is -3

Determine its 7th term.​

Answer :

7th term = -2187

Step-by-step explanation :

        $\underline{\underline{\bf Geometric \ Progression:}}$

• It is the sequence of numbers in which each number is obtained by multiplying the previous number by a constant.

• In GP,

          a - first term

          r - common ratio

         Tₙ - nth term

         Sₙ - sum of n terms

• General form of G.P.,

          a , ar , ar² , ar³ , ..... , arⁿ

• Formulae :-

         => nth term of G.P.,

              $\boxed{\bf T_n=ar^{n-1}}$

        => Sum of n terms.

            → r > 1

                $\boxed{\bf S_n=\frac{a(r^n-1)}{r-1} }$

            → r < 1

                 $\boxed{\bf S_n=\frac{a(1-r^n)}{1-r} }$

            → r = 1

                 $\boxed{\bf S_n=na}$

            → If GP has infinite terms,

                 $\boxed{\bf S_{\infty}=\frac{a}{1-r} }$

___________________________

Given,

• 4th term of a G.P. is square of its second term,

              T₄ = T₂²

• the first term = -3

              a = -3

So,

=> 4th term, T₄ = ar⁴⁻¹

                        = ar³

                        = -3r³

=> 2nd term, T₂ = ar²⁻¹

                          = ar¹

                          = -3r

According to the question,

        -3r³ = (-3r)²

        -3r³ = 9r²

         r³/r² = 9/-3

           r = -3

Common ratio, r = -3

$\underline{\bf 7^{th} \ term :}$

     = ar⁷⁻¹

     = ar⁶

     = (-3) (-3)⁶

     = (-3)⁷

     = -2187

∴ 7th term = -2187

Answer 2

Step-by-step explanation:

The 4" term of a G.P. is square of its second term, and the first term 15-3

Determine its 7th term.