Answer

Verified

64.2k+ views

Hint:-Here, this is a term of $AP$so we apply summation of $n$terms of$AP$.

Given series is\[2,3\dfrac{1}{4},4\dfrac{1}{2},\]…..Up to \[20\]terms.

The given series can be written as$\dfrac{8}{4},\dfrac{{13}}{4},\dfrac{{18}}{4}$,……to \[20\] terms.

Take $\dfrac{1}{4}$ common we get

$ \Rightarrow \dfrac{1}{4}(8 + 13 + 18 + ......)$ To \[20\] terms.

Observe that ${\text{8,13,18}}$…… is in Arithmetic Progression with first term $8$and$5$as common difference

So, the sum of first\[20\] terms in that series will be

$ = \dfrac{1}{4}\left( {\dfrac{{20}}{2}\left( {2 \times 8 + (20 - 1) \times 5} \right)} \right)$ $\because {S_n} = \dfrac{n}{2}\left( {2a + \left( {n - 1} \right)d} \right)$

Here $n = $ number of terms, $a = $ first term and $d = $ common difference.

${\text{ = }}\dfrac{1}{4}(1110) = \dfrac{{555}}{2}$ Answer.

Note: - Whenever such type of series is given first convert into simple form and then find which series is this. Then apply the formula of that series to get the answer.

Given series is\[2,3\dfrac{1}{4},4\dfrac{1}{2},\]…..Up to \[20\]terms.

The given series can be written as$\dfrac{8}{4},\dfrac{{13}}{4},\dfrac{{18}}{4}$,……to \[20\] terms.

Take $\dfrac{1}{4}$ common we get

$ \Rightarrow \dfrac{1}{4}(8 + 13 + 18 + ......)$ To \[20\] terms.

Observe that ${\text{8,13,18}}$…… is in Arithmetic Progression with first term $8$and$5$as common difference

So, the sum of first\[20\] terms in that series will be

$ = \dfrac{1}{4}\left( {\dfrac{{20}}{2}\left( {2 \times 8 + (20 - 1) \times 5} \right)} \right)$ $\because {S_n} = \dfrac{n}{2}\left( {2a + \left( {n - 1} \right)d} \right)$

Here $n = $ number of terms, $a = $ first term and $d = $ common difference.

${\text{ = }}\dfrac{1}{4}(1110) = \dfrac{{555}}{2}$ Answer.

Note: - Whenever such type of series is given first convert into simple form and then find which series is this. Then apply the formula of that series to get the answer.

Recently Updated Pages

Write a composition in approximately 450 500 words class 10 english JEE_Main

Arrange the sentences P Q R between S1 and S5 such class 10 english JEE_Main

What is the common property of the oxides CONO and class 10 chemistry JEE_Main

What happens when dilute hydrochloric acid is added class 10 chemistry JEE_Main

If four points A63B 35C4 2 and Dx3x are given in such class 10 maths JEE_Main

The area of square inscribed in a circle of diameter class 10 maths JEE_Main

Other Pages

For which reaction from the following Delta S will class 11 chemistry JEE_Main

What is the significance of the Heisenberg uncertainty class 11 chemistry JEE_Main

Electric field due to uniformly charged sphere class 12 physics JEE_Main

If a wire of resistance R is stretched to double of class 12 physics JEE_Main