Question

find the sum of indicated number of terms in each of the following arithmetic progresions 2,4,6,8,---100terms​

Answer 1

$\blue{\bold{\underline{\underline{Answer:}}}}$

$\green{\tt{\therefore{Sum\:of\:50th\:term=2550}}}$

$\orange{\bold{\underline{\underline{Step-by-step\:explanation:}}}}$

$\green{\underline \bold{Given:}} \\ \tt: { \implies First \: term(a_{1}) = 2 }\\ \\ \tt: {\implies Common \: difference(d) = 2}\\ \\ \tt: {\implies Last \: term(l) = 100} \\ \\ \red{\underline \bold{To \: Find:}} \\ \tt: {\implies Sum \: of \: n_{th} \: term = ?}$

• According to given question :

$\bold{As \: we \: know \:that} \\ \tt:{\implies a_{n} = a + (n - 1)d} \\ \\ \tt:{\implies 100 = 2 + (n - 1)2} \\ \\ \tt:{\implies 100 - 2 = 2n - 2} \\ \\ \tt:{\implies 98 + 2 = 2n} \\ \\ \tt:{\implies \frac{100}{2} = n }\\ \\ \tt: {\implies n = 50 \: terms } \\ \\ \bold{As \: we \: know \: that} \\ \tt:{\implies s_{n} = \frac{n}{2} (a + l) }\\ \\ \tt:{\implies s_{50} = \frac{50}{2} (2 + 100)} \\ \\ \tt: {\implies s_{50} = 25 \times 102} \\ \\ \green{\tt:{\implies s_{50} = 2550}}$

Answer 2

AnswEr :

$\bf{\green{\underline{\underline{\bf{Given\::}}}}}$

The terms in each of the following arithmetic progression 2,4,6,8.........100 terms.

$\bf{\green{\underline{\underline{\bf{To\:find\::}}}}}$

The sum of Indicated number.

$\bf{\green{\underline{\underline{\bf{Explanation\::}}}}}$

We know that $\sf{n^{th} }$ formula :

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

$\bf{We\:have}\begin{cases}\sf{First\:term\:(a)=2}\\ \sf{Common\:difference\:(d)=4-2=2}\\ \sf{Last\:term\:(l)=100}\end{cases}}$

A/q

$\mapsto\tt{100=2+(n-1)2}\\\\\\\mapsto\tt{100=\cancel{2}+2n\cancel{-2}}\\\\\\\mapsto\tt{100=2n}\\\\\\\mapsto\tt{n=\cancel{\dfrac{100}{2}}}\\ \\\\\mapsto\tt{\red{n=50}}$

Now, using formula of the sum :

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

$\mapsto\tt{S_{50}=\cancel{\dfrac{50}{2}} [2*2+(50-1)2]}\\\\\\\mapsto\tt{S_{50}=25(4+49*2)}\\\\\\\mapsto\tt{S_{50}=25(4+98)}\\\\\\\mapsto\tt{S_{50}=25(102)}\\\\\\\mapsto\tt{S_{50}=25\times 102}\\\\\\\mapsto\tt{\red{S_{50}=2550}}$

$\therefore\sf{\large{\pink{\underline{\sf{The\:sum\:is\:S_{50}=2550}}}}}$