Question

the 4th term of an AP is 22 and it's 10th term is 52,find the sum of its 40 terms?

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Answer 1

Step-by-step explanation:

t4: a+4d=22

t10: a+9d=52

on solving the above equation we get

a+9d=52

a+4d=22

- - -

0+5d=30

d=6

a= -2

Sn= n/2 [2a+(n-1)d]

S40= 40/2[2(-2)+(40-1)6]

S40= 20[-4+234]

S40= 20(230)

S40= 4600

Answer 2

$\large\textsf{ \textbf{\underline{Given :-}}}$

• 4th term of AP = 22
• 10th term of AP = 52

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$\large\textsf{ \textbf{\underline{To\:Find :-}}}$

• Sum of first 40 terms

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$\large\textsf{ \textbf{\underline{Solution :-}}}$

In order to find sum of 40 terms of AP, we have to firstly find out it's common difference.

Given that,

$\small:\longrightarrow \bf A_{4}=22$

nth term of AP can be expressed as :

• $\leadsto\bf A_n = A + ( n - 1 ) d$

Here :

• $\to \bf An = nth \:term$
• $\to\bf A = First\:term$
• $\to\bf d = Common\: difference$
• $\to\bf n = No.\:of\:terms$

So,

$\small :\longrightarrow \bf A+3d=22... \bf(1.)$

Given that,

$\small :\longrightarrow\bf A_{10}=52$

$\small:\longrightarrow \bf A+9d=52... \bf(2.)$

Substracting Eqn (1.) from (2.)

$\small:\longrightarrow\bf A+9d - (A+3d)=52 - 22$

$\small:\longrightarrow \bf A+9d - A - 3d=30$

$\small:\longrightarrow \bf 6d=30$

$\small:\longrightarrow\bf d= \dfrac{30}{5}$

$\small\bf :\longrightarrow d=5$

Put this value in Eqn (1.)

$\small\bf \leadsto A+3d=22$

$\small\bf \leadsto A+3(5)=22$

$\small\bf \leadsto A+15=22$

$\small\bf \leadsto A=22 - 15$

$\small\bf \leadsto A= 7$

So the first term of AP is 7.

Now we have formula for sum of AP,

$\displaystyle\underline{\boxed{\sf S_n=\dfrac{n}{2}[2A+(n-1)d]}}$

Here :

• Sn = Sum of n terms
• n = Number of terms
• A = First term
• d = Common difference

${\implies\bf S_{40}=\dfrac{40}{2} \big[2( A)+(40-1)(5) \big]}$

${\implies\bf S_{40}=20 \big[ 14+(39)(5) \big]}$

${\implies\bf S_{40}=20 \big[ 14+195 \big]}$

${\implies\bf S_{40}=20 \big[ 209 \big]}$

${\implies\bf S_{40}=4180}$

Hence the required sum of 40 terms of AP is 4180.

$\large\textsf{ \textbf{\underline{Additional\: Information:-}}}$

Shortcut trick to find common difference when 2 different terms or AP are given :-

• $\boxed{ \sf d = \dfrac{A_n -A_k }{n - k} }$

Here :

• Ak and An are two different terms of AP and d is the common difference.