Question

In an Ap, the sum of first 4 terms is 50 and sum of next four terms is 130. Find the sum of first 25 terms.

Answer 1

AnsweR :

$\bf{\Large{\underline{\sf{Given\::}}}}$

In an A.P., the sum of first 4 terms is 50 and sum of next four terms is 130.

$\bf{\Large{\underline{\sf{To\:find\::}}}}$

The sum of first 25 terms.

$\bf{\Large{\underline{\rm{\orange{Explanation\::}}}}}$

Formula use :

$\bf{\large{\boxed{\sf{Sum\:(S_{n})=\frac{n}{2} \big[2a+(n-1)d\big]}}}}}$

A/q

$\implies\tt{a_{1}+a_{2}+a_{3}+a_{4}=50}\\\\\\\\\implies\tt{\cancel{\dfrac{4}{2}} \big[2a+(4-1)d\big]=50}\\\\\\\\\implies\tt{2\big[2a+3d\big]=50}\\\\\\\\\implies\stt{2a+3d=\cancel{\dfrac{50}{2}} }\\\\\\\\\implies\tt{\blue{2a+3d=25..........................(1)}}$

&

$\implies\tt{a_{5}+a_{6}+a_{7}+a_{8}=130}\\\\\\\\\implies\tt{S_{8}-S_{4}=130}\\\\\\\\\implies\tt{\cancel{\dfrac{8}{2} }\big[2a+(8-1)d\big]-50=130}\\\\\\\\\implies\tt{4\big[2a+7d\big]-50=130}\\\\\\\\\implies\tt{4\big[2a+7d\big]=130+50}\\\\\\\\\implies\tt{4\big[2a+7d\big]=180}\\\\\\\\\implies\tt{2a+7d=\cancel{\dfrac{180}{4} }}\\\\\\\\\implies\tt{\blue{2a+7d=45........................(2)}}$

$\bigstar$ Subtracting equation (1) from equation (2), we get;

$\leadsto\sf{\cancel{2a}+7d\cancel{-2a}-3d=45-25}\\\\\\\leadsto\sf{7d-3d=20}\\\\\\\leadsto\sf{4d\:=\:20}\\\\\\\leadsto\sf{d\:=\:\cancel{\dfrac{20}{4} }}\\\\\\\leadsto\sf{\blue{d\:=\:5}}$

$\bigstar$ Putting the value of d in equation (1), we get;

$\leadsto\tt{2a+3(5)=25}\\\\\\\leadsto\tt{2a+15=25}\\\\\\\leadsto\tt{2a=25-15}\\\\\\\leadsto\tt{2a=10}\\\\\\\leadsto\tt{a\:=\:\cancel{\dfrac{10}{2} }}\\\\\\\leadsto\tt{\blue{a\:=\:5}}$

Now,

$\implies\tt{S_{25}=\dfrac{25}{2} \big[2(5)+(25-1)*5\big]}\\\\\\\\\implies\tt{S_{25}=\dfrac{25}{2} \big[10+24*5\big]}\\\\\\\\\implies\tt{S_{25}=\dfrac{25}{2} \big[10+120\big]}\\\\\\\\\implies\tt{S_{25}=\dfrac{25}{\cancel{2}} *\cancel{130}}\\\\\\\\\implies\tt{S_{25}=(25*65)}\\\\\\\\\implies\tt{\blue{S_{25}\:=\:1625}}$

Thus,

$\bigstar$The sum of first 25 terms is 1625.

Answer 2

$\huge \boxed {\fcolorbox{cyan}{red}{Answer: }}$

Given question..

In an Ap, the sum of first 4 terms is 50 and sum of next four terms is 130. Find the sum of first 25 terms.

Answer

given ap is the sum of first 4 term is 50

sum of next four term is 130

$\sf{so \: now \: first \: term \: 25 \: terms}$

$\sf{a1 + a2 + a3 + a4 = 50}$

$\sf{ \frac{4}{2} \: [2a + (4 - 1)d] = 50}$

$\sf{2[2a + 3d] = 50}$

then

$\sf{2a + 3d = \frac{50}{2}}$

so..

$\sf{2a + 3d = 25 \: is \: equation \: one}$

$\sf{given \: 130}$

$\sf{ a5 + a6 + a7 + a8 = 130}$

then

$\sf{s3 + s4 = 130}$

$\sf{ \frac{8}{2}[2a + (8 - 1)d] - 50 = 130}$

$\sf{4[2a + 7d] = 130 + 50}$

$\sf{4[2a + 7d] = 180}$

$\sf{2a + 7d = \frac{180}{4}}$

$\sf{2a + 7d = 45 \: is \: eq \: 2}$

then subtracting equation 2 we get

$\rm{2a + 7d - 2a - 3d = 45 - 25}$

$\rm{7d - 3d =20}$

$\rm{4d = 20}$

$\rm{d = \frac{20}{4}}$

$\rm{d = 5}$

so now put the value of d in eq 1 we get

$\rm{2a + 3(5) = 25}$

$\rm{2a + 15 = 25}$

$\rm{2a = 25 - 15}$

$\rm{2a = 10}$

$\rm{a = \frac{10}{2}}$

$\sf{a = 5}$

then now use the formula used

$\sf{s25 = \frac{25}{2}[2(5) + (25 - 1) \times 5]}$

$\sf{s25 = \frac{25}{2}[10 + 24 \times 5]}$

$\sf{s25 = \frac{25}{2}[10 + 120]}$

$\sf{s25 = \frac{25}{2} \times 130}$

$\sf{s25 = (25 \times 65)}$

$\sf{s25 = 1625}$

$\bf{ \huge{ \boxed{ \green{ \tt{sum \:of \: frst \: 25 \: term \: is \: 1625}}}}}$