Ncert Solutions Maths class 11th

79. An A.P. of numbers from 1 to 100 divisible by 2 is

2, 4, 6, ……….98, 100.

So, a = 2 and d = 4-2 = 2

$l=$ 100

$\Rightarrow a+(n-1)d=100$

$\Rightarrow 2+(n-1)2=100$

$\Rightarrow \frac{2}{2}+(n-1)\frac{2}{2}=\frac{100}{2}$

$\Rightarrow 1+(n-1)=50$

$\Rightarrow n=50$

Let, $s_1=2+4+6?+9+100=\frac{50}{2}[2+100]\left\{?S_n=\frac{n}{2}(a+1)\right\}$

$=\frac{50*102}{2}$

= 2550

Similarly, an A.P. of numbers from 1 to 100 divisible by 5 is

5,10,15, ……. 95,100

So, a = 5 and d = 100

$l=$ 100

$\Rightarrow a+(n-1)d=100$

$\Rightarrow 5+(n-1)5=100$

$\Rightarrow 1+(n-1)=20$

$\Rightarrow n=20$

$\therefore S_2=5+10+15\dots +95+100=\frac{20}{2}[5+100]$

$=\frac{20*105}{2}$

= 1050

As there are also no divisible by both 2 and 5 , i.e., LCM of 2 and 5 = 10 An A.P. of no. from 1 to 100 divisible by 10 is 10, 20, …………100

So, a = 10, d = 10

$l=100$

$\Rightarrow a+(x-1)l=100$

$\Rightarrow 10+(x-1)10=100$

$\Rightarrow 1+(x-1)=10$

$\Rightarrow x=10$

So, $S_3=10+20+30+\dots +100=\frac{10}{2}[10+100]$

= 550

The required sum of number $=S_1+S_2{S}_3=2550+1050-550=3050$

= 3050

78. $\begin{array}{ccccc}\hfill 7\frac{28}{200}\hfill & \hfill \frac{57}{400}\hfill & \hfill \frac{14}{\frac{60}{\frac{56}{4}}}\hfill & \hfill \frac{35}{\begin{array}{l}50\\ \frac{49}{1}\\ \end{array}}\hfill & \hfill \hfill \\ \hfill \hfill & \hfill Smallest =200-4+7=203\hfill & \hfill \hfill & \hfill Largest =400-1=399\hfill & \hfill \hfill \end{array}$

So we can form an A.P. 203,203+7, …., 399-7, 399

So, i.e. a = 203 and d = 7 $$

l = 399

$\Rightarrow a+(x-1)d=399$

$\Rightarrow 203+(x-1)7=399$

$\Rightarrow (x-1)7=399-203$

$\Rightarrow x-1=\frac{196}{7}$

$\Rightarrow x=28+1$

$\Rightarrow x=29$

Sum of the 29 number of the AP = Required sum = $\frac{\eta }{2}[a+l)$

$=\frac{29}{2}[203+399]$

$=\frac{29}{2}*602$

= 8729.

77. Let a and d be the first term and common difference of an A.P.

Then, 

$n_{}=\frac{n}{2}[2a+(n-1)d]=S_1$

$S_{2n}=\frac{2n}{2}[2a+(2n-1)d]=S_2$

$S_{3n}=\frac{3n}{2}[2a+(3n-1)d]=S_3$

Now, R.H.S $=3\left(S_2-S_1\right)$

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

$=\frac{3n}{2}\{2[2a+(2n-1)d-[2a+(x-1)d]\}$

$=\frac{3n}{2}\{4a+(4x-2)d-2a-(x-1)d\}$

$=\frac{3n}{2}\{2a+\left[4x-2\left(n-1\right)d\right]\}$

$=\frac{3n}{2}\{2a+[4n-2-n+1]d\}$

$=\frac{3n}{2}\{2a+[3n-1]d\}=S_3$

$\therefore S_3=3\left(S_2-s_1\right)$

76. Let the three numbers a $-$ d, a, a + d be in A.P.

Then, $\left(a-d\right)+a+\left(a+d\right)=24$

$\Rightarrow$ 3a = 24

$\Rightarrow$ a = 8

and, $(a-d)*a*(a+d)=440$

$\Rightarrow (a-d)\left(a^2+ad\right)=440$

$\Rightarrow a^3+a^{2}d-a^{2}d-a{d}^2=440$

$\Rightarrow a^3-a{d}^2=440.$

$\Rightarrow a\left(a^2-d^2\right)=440$

Put, $a=8,\Rightarrow 8\left(a^2-d^2\right)=440$

$\Rightarrow 64-d^2=\frac{440}{8}$

$\Rightarrow 64-d^2=55$

$d^2=64-55$

$\Rightarrow d^2=9$

$\Rightarrow d=\pm 3$

When d = 3, a = 8 the three number are.

8 $-$ 3, 8, 8 + 3 5,8,11

When d = $-$ 3, a = 8 the three numbers are.

$8-(-3),8,8+(-3)\Rightarrow 8+3,8,8-3\Rightarrow 11,8,5$

75. Let a and d be the first term and common difference of the A.P.

So,  $a(m+n)+a(m-n)=\{a+[(m+n)-1]d\}+\{a+[[m-n)-1]d\}$

$=a+\left(m+n-1\right)d+a+\left(m-n-1\right)d$

$=2a+\left(m+n-1+m-n-1\right)d$

$=2a+\left(2m+0-2\right)d$

$=2[a+(m-1)d]$

= 2 am

72. Given that,

a_n = n(n + 1)(n + 4)

= n(n² + 4n + n + 4)

=n(n² + 5n + 4)

= x³ + 5x² + 4x

So, sum of terms, S_n = $\sum n^3+5\sum n^2+4{\displaystyle \sum _{}^{}n}$

$={\left[\frac{n(n+1)}{2}\right]}^2+5*\frac{n(n+1)(2n+1)}{6}+\frac{4\stackrel{?}{n}(n+1)}{2}$

$=\frac{n(n+1)}{2}\left[\frac{n(n+1)}{2}+\frac{5(2n+1)}{3}+4\right]$

$=\frac{n(n+1)}{2}\left[\frac{3n(n+1)+2*5(2n+1)+6*4}{6}\right].$

$=\frac{n(n+1)}{2}\left[\frac{3n^2+3n+20n+10+24}{6}\right]$

$=\frac{n(n+1)}{2}\left[\frac{3n^2+23n+34}{6}\right]$

$=\frac{n(n+1)\left(3n^2+23n+34\right)}{12}$

$=\frac{n(n+1)}{12}$ $\left(3n^2+6n+17n+34\right)$

$=\frac{n(n+1)}{12}[3n^2+23n+34]$

71. The given series is 1² + (1² + 2²) + (1² + 2² + 3²) + …

So, nth term well be

a_n =1² + 2² + 3² + … + n².

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

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

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

$=\frac{2n^3+3n^2+n}{6}$

$=\frac{x^3}{3}+\frac{x^2}{2}+\frac{x}{6}$

So, S_n = $\frac{1}{3}\sum n^3+\frac{1}{2}\sum n^2+\frac{1}{6}{\displaystyle \sum _{i=1}^{n}n}$

$=\frac{1}{3}{n}^2\frac{{(n+1)}^2}{4}+\frac{1}{2}\frac{n(n+1)(2n+1)}{6}+\frac{1}{6}·\frac{n(n+1)}{2}$

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

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

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

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

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

70. The given series is 3 8 + 11 + 9 14 + …

So, a_n= (nth term of 3, 6, 9, …) (nth term of 8, 11, 14, …)

i e, a = 3, d = 6- 3 = 3i e, a= 8, d = 11- 8 = 3

= [3 + (n- 1) 3] [8 + (n -1) 3]

= [3 + 3n- 3] [8 + 3n -3]

= 3n (3n + 5)

= 9n² + 15n.

So, S_n = 9∑n² + 15∑ n.

$=9*\frac{n(n+1)(2n+1)}{6}+15*\frac{n(n+1)}{2}$

$=\frac{3}{2}n(n+1)(2n+1)+\frac{15}{2}n(n+1)$

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

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

= 3n (n + 1) (n + 3)

69. The given series is 5² + 6² + 7² + … + 20²

This can be rewritten as (1² +2² + 3² + 4² + 5² + 6² + 7² + … + 20²) - (1² + 2² + 3² + 4²)

So, sum = (1² + 2² + 3² + 4² … + 20²) - (1² + 2²+ 3² + 4²)

$=$${\displaystyle \underset{}{\overset{}{}}}$${\displaystyle \sum _{i=1}^{20}{n}^2}-{\displaystyle \sum _{i=1}^4{n}^2}$

$=\frac{20(20+1)(2*20+1)}{6}-\frac{4(4+1)(4*2+1)}{6}$

$=\frac{20*21*41}{6}-\frac{4*5*9}{6}$

= 2870 30

= 2840.

68. The given series is $\frac{1}{1*2}+\frac{1}{2*3}+\frac{1}{3*4}+?$

So, an = $\frac{1}{\left(n^{}1,2,3\dots \right)}*\frac{1}{\left(n^{ }ten 2,3,4\dots \right)}$

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

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

$=\frac{1}{n(n+1)}$

$=\frac{(n+1)-n}{n(n+1)}$

$=\frac{(n+1)}{n(n+1)}-\frac{n}{n(n+1)}$

$a_n=\frac{1}{n}-\frac{1}{n+1}$

So. Putting n = 1, 2, 3….n.

a₁ = $\frac{1}{1}-\frac{1}{2}$

a₂ = $\frac{1}{2}-\frac{1}{3}$

a₃ = $\frac{1}{3}-\frac{1}{4}\dots .$

So, adding. L.H.S and R.H.S. up ton terms

a₁ + a² + a₃ + … + a_n = $\left[\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+?\frac{1}{n}\right]-[\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+?·\frac{1}{n}+\frac{1}{n+1}]$

 S_n = 1 $\frac{1}{n+1}$ { equal terns cancelled out}

 S_n = $\frac{n+1-1}{.n-1}$

 S_n = $\frac{n}{n+1}$