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Maksim231197 [3]
2 years ago
8

The sum of the series {1(2/3)}²+{2(1)3)}²+3²+{3(2/3)}²+....to 10 term is

Mathematics
1 answer:
ryzh [129]2 years ago
6 0

Step-by-step explanation:

<h3><u>Given Question </u></h3>

The sum of the series is

\tt{ {\bigg[1\dfrac{2}{3} \bigg]}^{2} + {\bigg[2\dfrac{1}{3} \bigg]}^{2}  +  {3}^{2} + {\bigg[3\dfrac{2}{3} \bigg]}^{2} +  -  -  10 \: terms}

\green{\begin{gathered}\large{\sf{{\underline{Formula \: Used - }}}}  \end{gathered}}

\boxed{\tt{ \displaystyle\sum_{k=1}^{n}1 = n \: }}

\boxed{\tt{ \displaystyle\sum_{k=1}^{n}k =  \frac{n(n + 1)}{2}  \: }}

\boxed{\tt{ \displaystyle\sum_{k=1}^{n} {k}^{2}  =  \frac{n(n + 1)(2n + 1)}{6}  \: }}

\large\underline{\sf{Solution-}}

Given series is

\rm :\longmapsto\: {\bigg[1\dfrac{2}{3} \bigg]}^{2} + {\bigg[2\dfrac{1}{3} \bigg]}^{2}  +  {3}^{2} + {\bigg[3\dfrac{2}{3} \bigg]}^{2} +  -  -  - 10 \: terms

can be rewritten as

\rm \:  =  \:  {\bigg[\dfrac{5}{3} \bigg]}^{2} + {\bigg[\dfrac{7}{3} \bigg]}^{2} + {\bigg[\dfrac{9}{3} \bigg]}^{2} +   {\bigg[\dfrac{11}{3} \bigg]}^{2} +  -  -  - 10 \: terms

\rm \:  =  \: \dfrac{1}{9}[ {5}^{2} +  {7}^{2} +  {9}^{2} +  -  -  - 10 \: terms \: ]

Now, here, 5, 7, 9 forms an AP series with first term 5 and common difference 2.

So, its general term is given by 5 + ( n - 1 )2 = 5 + 2n - 2 = 2n + 3

So, above series can be represented as

\rm \:  =  \: \dfrac{1}{9}\displaystyle\sum_{n=1}^{10}(2n + 3) ^{2}

\rm \:  =  \: \dfrac{1}{9}\displaystyle\sum_{n=1}^{10}\bigg[ {4n}^{2} + 9 + 12n\bigg]

\rm \:  =  \: \dfrac{1}{9}\bigg[\displaystyle\sum_{n=1}^{10} {4n}^{2} + \displaystyle\sum_{n=1}^{10}9 + 12\displaystyle\sum_{n=1}^{10}n\bigg]

\rm \:  =  \: \dfrac{1}{9}\bigg[4\displaystyle\sum_{n=1}^{10} {n}^{2} +9 \displaystyle\sum_{n=1}^{10}1 + 12\displaystyle\sum_{n=1}^{10}n\bigg]

\rm \:  =  \: \dfrac{4}{9}\bigg[\dfrac{10(10 + 1)(20 + 1)}{6} \bigg] + 10 + \dfrac{4}{3}\bigg[\dfrac{10(10 + 1)}{2} \bigg]

\rm \:  =  \: \dfrac{4}{9}\bigg[\dfrac{10(11)(21)}{6} \bigg] + 10 + \dfrac{4}{3}\bigg[\dfrac{10(11)}{2} \bigg]

\rm \:  =  \: \dfrac{1540}{9}  + 10 + \dfrac{220}{3}

\rm \:  =  \: \dfrac{1540 + 90 + 660}{9}

\rm \:  =  \: \dfrac{2290}{9}

Hence,

\boxed{\tt{ {\bigg[1\dfrac{2}{3} \bigg]}^{2} + {\bigg[2\dfrac{1}{3} \bigg]}^{2}  +  {3}^{2} + {\bigg[3\dfrac{2}{3} \bigg]}^{2} +  -  -  10 \: terms =  \frac{2290}{9}}}

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