Math, asked by kjanehitalada, 6 months ago

the number of terms of the series 5+13+ ... that must be added so that the sum will be 495​

Answers

Answered by Anonymous
9

Given :

  • First term = 5

To find :

No. of terms of the AP.

Solution :

Common difference of the AP :-

We know the formula for common difference of the AP i.e,

\boxed{\bf{d = a_{n} - a_{(n - 1)}}}

Where :

  • d = Common Difference

  • a = Any term of the AP.

Here ,

⠀⠀⠀⠀⠀⠀⠀⠀⠀a(n -1) = 1st term

⠀⠀⠀⠀⠀⠀⠀⠀⠀an = 2nd term

Now using the formula for common difference and substituting the values in it, we get :

:\implies \bf{d = a_{n} - a_{(n - 1)}} \\ \\ \\

:\implies \bf{d = 13 - 5} \\ \\ \\

:\implies \bf{d = 8} \\ \\ \\

\boxed{\therefore \bf{d = 8}} \\ \\ \\

Hence the common difference of the AP is 8.

No. of terms of the AP :

We know the formula for sum of n terms of the AP i.e,

\boxed{\bf{s_{n} = \dfrac{n}{2}\bigg(2a_{1} + (n - 1)d\bigg)}}

Where :

  • s = Sum of terms of the AP.

  • n = No. of terms

  • d = Common Difference

  • a = First term

Now using the formula for sum of n terms and substituting the values in it, we get :

:\implies \bf{s_{n} = \dfrac{n}{2}\bigg(2a_{1} + (n - 1)d\bigg)} \\ \\ \\

:\implies \bf{495 = \dfrac{n}{2}\bigg(2 \times 5 + (n - 1)8\bigg)} \\ \\ \\

:\implies \bf{495 = \dfrac{n}{2}\bigg(10 + 8n - 8\bigg)} \\ \\ \\

:\implies \bf{495 = \dfrac{10n + 8n^{2} - 8n}{2}} \\ \\ \\

:\implies \bf{495 \times 2 = 10n + 8n^{2} - 8n} \\ \\ \\

:\implies \bf{495 \times 2 = 8n^{2} + 2n} \\ \\ \\

:\implies \bf{990  = 8n^{2} + 2n} \\ \\ \\

:\implies \bf{8n^{2} + 2n - 990 = 0} \\ \\ \\

:\implies \bf{8n^{2} + (90 - 88)n - 990 = 0} \\ \\ \\

:\implies \bf{8n^{2} - 88n + 90n - 990 = 0} \\ \\ \\

:\implies \bf{8n(n - 11) + 90(n - 11) = 0} \\ \\ \\

:\implies \bf{(n - 11)(8n + 90) = 0} \\ \\ \\

:\implies \bf{(n - 11 = 0)\quad(8n + 90 = 0)} \\ \\ \\

:\implies \bf{(n = 11)\quad(8n = -90) } \\ \\ \\

:\implies \bf{n = 11\quad n = \dfrac{-90}{8}} \\ \\ \\

\boxed{\therefore \bf{n = 11,\dfrac{-90}{8}}} \\ \\ \\

Since the no. of terms cannot be negative and in fraction , the actual number of terms of the AP is 11. (n = 11)

Hence their are 11 terms in the AP.

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