Sequences and Series

100. Given, $\frac{1*2^2+2*3^2+?+n{(n+1)}^2}{1^2*2+2^2+3+?+n^2(n+1)}$

For numerator,

a  (n^th term) = n (n + 1)² = n (n + 1)² = n (n² + 2n +1) = n³ + 2n² + n

So, $S_n={\displaystyle \sum _{}^{}{a}_n}={\displaystyle \sum _{}^{}{n}^3}+2{\displaystyle \sum _{}^{}{n}^2}+{\displaystyle \sum _{}^{}n}$

$=\frac{n^2(n+1){}^2}{4}+\frac{2\cdot (n+1)(2n+1)n}{6}+\frac{n(n+1)}{2}$

$=\frac{n(n+1)}{2}\left[\frac{n(n+1)}{2}+\frac{2(2n+1)}{3}+1\right]$

$=\frac{n(n+1)}{2}\left[\frac{3n(n+1)+2*2(2n+1)+6}{6}\right]$

$=\frac{n(n+1)}{12}\left[3n^2+3n+8n+4n+6\right]$

$=\frac{n(n+1)}{12}\left[3n^2+11n+10\right]$

$=\frac{n(n+1)}{12}\left[3n^2+6n+5n+10\right]$

$=\frac{n(n+1)}{12}[(33n(n+2)+5(n+2)]$

$=\frac{n(n+1)(n+2)(3n+5)}{12}$

For denominator,

a_n (n^th term) = n² (n + 1) = n³ + n²

So,

$S_n={\displaystyle \sum _{}^{}{a}_n}={\displaystyle \sum _{}^{}{n}^3}+{\displaystyle \sum _{}^{}{n}^2}$

$=\frac{n^2(n+1){}^2}{4}+\frac{n(n+1)(2n+1)}{6}$

$=\frac{n(n+1)}{2}\left[\frac{n(n+1)}{2}+\frac{(2n+1)}{3}\right]$

$=\frac{n(n+1)}{2}\left[\frac{3n(n+1)+2(2n+1)}{6}\right]$

$=\frac{n(n+1)}{2}\left[\frac{3n^2+3n+4n+2}{6}\right]$

$=\frac{n(n+1)}{2}\left[\frac{3n^2+6n+1n+2}{6}\right]$

$=\frac{n(n+1)}{2}\left[\frac{3n(n+2)+(n+2)}{6}\right]$

$=\frac{n(n+1)(n+2)(3n+1)}{12}]$

So, $\frac{1*2^2+2*3^2+?+n{(n+1)}^2}{1^2*2+2^2*3+?+n^2(n+1)}=\frac{\frac{n(n+1)(n+2)(3n+5)}{12}}{\frac{n(n+1)(n+2)(3n+1)}{12}}$

$=\frac{3n+5}{3n+1}$

99. The given series is.

$\frac{1^3}{1}+\frac{1^3+2^3}{1+3}+\frac{1^3+2^3+3^3}{1+3+5}$ +…………. upto nth term,

The nth term is ,

$a_n=\frac{1^3+2^3+3^3+........+n^3}{1+3+5+........+n...terms}$ { A.P. of n terms and a = 1, d = 5-3 = 2}

$a_n=\frac{{\left[\frac{n(n+1)}{2}\right]}^2}{\frac{n}{2}[2*1+(n-1)2]}$

$a_n=\frac{\frac{n^2(n+1){}^2}{4}}{\frac{n}{2}[2+(n-1)2]}=\frac{\frac{n^2(n+1){}^2}{4}}{n[1+n-1]}$

$a_n=\frac{\frac{n^2(n+1){}^2}{4}}{n^2}=\frac{{(n+1)}^2}{4}$

$a_n=\frac{n^2+2x+1}{4}=\frac{n^2}{4}+\frac{n}{2}+\frac{1}{4}$

$\therefore Sn={\displaystyle \sum _{}^{}an}=\frac{1}{4}{\displaystyle \sum _{}^{}{x}^2}+\frac{1}{2}{\displaystyle \sum _{}^{}n+{\displaystyle \sum _{}^{}1}}$

$=\frac{1}{4}*\frac{9(n+1)(2n+1)}{6}+\frac{1}{2}*\frac{n(x+1)}{2}+\frac{1}{4}*n$

$=\frac{n}{4}\left[\frac{(n+1)(2n+1)}{6}+(n+1)+1\right]$

$=\frac{n}{4}\left[\frac{2n^2+n(n+1)(2n+1)+6(x+1)+6}{6}\right]$

$=\frac{n}{4}\left[\frac{2n^2+n+2n+1+6n+6+6}{6}\right]$

$=\frac{n}{4}\left[\frac{2n^2+9n+13}{6}\right]$

$=\frac{n\left(2n^2+9n+13\right)}{24}$

98. Given,

S₁ = 1 + 2 + 3 + ………. + n $=n+\frac{(n+1)}{2}$

S₂ = 1² + 2² + 3² + ………… + n²  $=\frac{n(n+1)(2n+1)}{6}$

S₃ = 1³ + 2³ + 3³ + …………. + n³ =${\left[\frac{n(n+1)}{2}\right]}^2$

So, L.H.S. $=9S_2^2=9*{\left[\frac{n(n+1)(2n+1)}{6}\right]}^2$

$=\frac{9*n^2(n+1){}^2(2n+1){}^2}{36}$

$=\frac{n^2(n+1){}^2(2n+1){}^2}{4}$

R.H.S. $=S_3\left(1+8S_1\right)={\left[\frac{n(n+1)}{2}\right]}^2\left[1+8*\frac{n(n+1)}{2}\right]$

$=\frac{n^2(n+1){}^2}{4}*[1+4n(n+1)]$

$=\frac{n^2(n+1){}^2}{4}\left[1+4n^2+4n\right]$

$=\frac{n^2(n+1){}^2}{4}\left[{(2n)}^2+2(2n)1+1^2\right]$

$=\frac{n^2(n+1){}^2}{4}{(2n+1)}^2=$ L.H.S.

So, 9(S₂)² = S₃ ( 1 + 8S₁)

97. The given series is 3+7+13+21+31+ ……. upto n terms

So, Sum, S_n = 3+7+13+21+31+ ………….+ a_n-1 + a_n

Now, taking,

S_n = S_n = [ 3 + 7 + 13 + 21 + 31 + ………. a_n-1 + a_n ] [ 3+ 7 + 21 + 31 + … a_n-1 + a_n]

0 = [ 3+ (7 - 3) + (13 - 7) + (13 - 7) + (21 - 13) + …….+ (a_n a_n-1) - a_n]

0 = 3 + [ 4 + 6 + 8 +…….+ (n-1) terms] a_n

$\Rightarrow a_n=3+\left\{\frac{n-1}{2}[2*4+[(n-1)-1]2]\right\}$ { $?$ 4 + 6 + 8 + ……. is sum of A.P. of n 1 terms with a = 4, d = 6- 4 = 2}

$\Rightarrow a_n=3+\left\{\frac{n-1}{2}[8+(n-2)2]\right\}$

$\Rightarrow a_n=3+\left\{\frac{n-1}{2}*2[4+n-2]\right\}=3+(n-1)(n+2)$

a_n = 3 + n² + (1 + 2) n + (-1) (2) { $?$ (a + b) (a + b) = a2 + (b + c) a + bc }

a_n = 3 + n² + n- 2 = n² + n +1

sum of series, $S_n={\displaystyle \sum _{}^{}{a}_n}={\displaystyle \sum _{}^{}{n}^2}+{\displaystyle \sum _{}^{}n}+{\displaystyle \sum _{}^{}1}$

$=\frac{n(n+1)(2n+1)}{6}+\frac{n(x+1)}{2}+n$

$=n\left[\frac{(n+1){(2n+1)}^{}}{6}+\frac{n+1}{2}+1\right]$

$=n\left[\frac{(n+1)(2n+1)+3(n+1)+6}{6}\right]$

$=n\left[\frac{2n^2+n+2n+1+3n+3+6}{6}\right]$

$=n\left[\frac{2n^2+6n+10]}{6}\right]$

$=n\left[\frac{2\left(n^2+3n+5\right)}{6}\right]$

$=\frac{n\left(n^2+3n+5\right)}{3}$

96. Given, series is $2*4+4*6+6*8+\dots +n$ terms.

So, a₂₀ = ( 20th term of A.P. 2, 4, 6 ….) ( 20th term of A.P. 4,6,8 ………)

i.e. a = 2, d = 4 -2 = 2 i.e. a =4, d = 6- 4 = 2

= [2 + (20- 1)2] [4 + (20-1)2]

= [2+19*2] [4 +19 *2]

= (2+38) (4+38)

= 40*42 = 1680

95. (i) Sn = 5+55+555+…………… upto n term

=5(1+11+111+ …………….upto n term )

Multiplying and dividing by 9 we get

$=\frac{5}{9}(9+99+999+\dots upto\mathrm{nterms})$

 

$=\frac{5}{9}\left[\right(10+10^2+10^3+?$ upto\mathrm{nterms}$$
$-n*1]$

$=\frac{5}{9}\left[\frac{10\left(10^n-1\right)}{10-1}-n\right]$ { $?10+10^2+10^3+\dots \mathrm{ntermsisaG.Pof}a=10,r=10>1\mathrm{nterms}$ }

$=\frac{5}{9}*\left[\frac{10}{9}\left(10^n-1\right)-n\right]$

$=\frac{50}{81}\left(10^n-1\right)-\frac{5n}{9}$

(ii) $S_n=0.6+0.66+0.666+\dots n$

$=6[0.1+0.11+0.111+?uptonterms$$]$

$=\frac{6}{9}[0.9+0\cdot 99+0.999+?\mathrm{<math><mrow><mo>uptonterms</mo></mrow></math>}]$ {multiplying and dividing by 9}

$=\frac{6}{9}[(1+1+1\dots n)-(0.1+0.01+0.001+\dots \mathrm{<math><mrow><mo>uptonterms</mo></mrow></math>})]$

$=\frac{6}{9}\left[n-\left(\frac{1}{10}+\frac{1}{10}*\frac{1}{10}+\frac{1}{10}*{\left(\frac{1}{10}\right)}^2+?\mathrm{<math><mrow><mo>uptonterms</mo></mrow></math>}\right)\right]$

$=\frac{6}{9}\left[n-\frac{\frac{1}{10}\left(1-{\left(\frac{1}{10}\right)}^n\right)}{1-\frac{1}{10}}\right]$

$=\frac{6}{9}\left[n-\frac{1-{\left(\frac{1}{10}\right)}^n}{10-1}\right]$

$=\frac{6}{9}\left[x-\frac{1}{9}\left(1-\frac{1}{10^n}\right)\right]$

$=\frac{6}{9}*\frac{1}{9}\left[9x-\left(1-\frac{1}{10^n}\right)\right]$

$=\frac{2}{27}\left[9n-\left(1-\frac{1}{10^n}\right)\right]$

$=\frac{2}{27}\left[9n-1+\frac{1}{10^n}\right]$