Class 10: Arithmetic Progressions

$\displaystyle \textbf{Question 1: }\text{Write the first five terms of each of the following sequences whose }n\text{th terms are:}$
$\displaystyle \text{(i) }a_n=3n+2\qquad\text{(ii) }a_n=3^n\qquad\text{(iii) }a_n=(-1)^n\cdot2$
$\displaystyle \text{(iv) }a_n=\frac{n(n-2)}{2}\qquad\text{(v) }a_n=n^2-n+1\qquad\text{(vi) }a_n=2n^2-3n+1$
$\displaystyle \text{Answer:}$
$\displaystyle \text{(i) Given, }a_n=3n+2$
$\displaystyle a_1=3(1)+2=5,\quad a_2=3(2)+2=8,\quad a_3=3(3)+2=11$
$\displaystyle a_4=3(4)+2=14,\quad a_5=3(5)+2=17$
$\displaystyle \therefore \text{The first five terms are }5,8,11,14,17.$

$\displaystyle \text{(ii) Given, }a_n=3^n$
$\displaystyle a_1=3^1=3,\quad a_2=3^2=9,\quad a_3=3^3=27$
$\displaystyle a_4=3^4=81,\quad a_5=3^5=243$
$\displaystyle \therefore \text{The first five terms are }3,9,27,81,243.$

$\displaystyle \text{(iii) Given, }a_n=(-1)^n\cdot2^n$
$\displaystyle a_1=(-1)^1\cdot2^1=-2,\quad a_2=(-1)^2\cdot2^2=4,\quad a_3=(-1)^3\cdot2^3=-8$
$\displaystyle a_4=(-1)^4\cdot2^4=16,\quad a_5=(-1)^5\cdot2^5=-32$
$\displaystyle \therefore \text{The first five terms are }-2,4,-8,16,-32.$

$\displaystyle \text{(iv) Given, }a_n=\frac{n(n-2)}{2}$
$\displaystyle a_1=\frac{1(1-2)}{2}=-\frac12,\quad a_2=\frac{2(2-2)}{2}=0,\quad a_3=\frac{3(3-2)}{2}=\frac32$
$\displaystyle a_4=\frac{4(4-2)}{2}=4,\quad a_5=\frac{5(5-2)}{2}=\frac{15}{2}$
$\displaystyle \therefore \text{The first five terms are }-\frac12,0,\frac32,4,\frac{15}{2}.$

$\displaystyle \text{(v) Given, }a_n=n^2-n+1$
$\displaystyle a_1=1^2-1+1=1,\quad a_2=2^2-2+1=3,\quad a_3=3^2-3+1=7$
$\displaystyle a_4=4^2-4+1=13,\quad a_5=5^2-5+1=21$
$\displaystyle \therefore \text{The first five terms are }1,3,7,13,21.$

$\displaystyle \text{(vi) Given, }a_n=2n^2-3n+1$
$\displaystyle a_1=2(1)^2-3(1)+1=0,\quad a_2=2(2)^2-3(2)+1=3$
$\displaystyle a_3=2(3)^2-3(3)+1=10,\quad a_4=2(4)^2-3(4)+1=21$
$\displaystyle a_5=2(5)^2-3(5)+1=36$
$\displaystyle \therefore \text{The first five terms are }0,3,10,21,36.$
$\displaystyle \\$

$\displaystyle \textbf{Question 2: }\text{Find the indicated terms in each of the following sequences whose }n\text{th terms are:}$
$\displaystyle \text{(i) }a_n=5n-4;\ a_{12}\text{ and }a_{15}\qquad\text{(ii) }a_n=\frac{3n-2}{4n+5};\ a_7\text{ and }a_8$
$\displaystyle \text{(iii) }a_n=n(n-1)(n-2);\ a_5\text{ and }a_8\qquad\text{(iv) }a_n=(-1)^n n;\ a_3,a_5,a_8$
$\displaystyle \text{Answer:}$
$\displaystyle \text{(i) Given, }a_n=5n-4$
$\displaystyle a_{12}=5(12)-4=60-4=56$
$\displaystyle a_{15}=5(15)-4=75-4=71$
$\displaystyle \therefore a_{12}=56\text{ and }a_{15}=71.$

$\displaystyle \text{(ii) Given, }a_n=\frac{3n-2}{4n+5}$
$\displaystyle a_7=\frac{3(7)-2}{4(7)+5}=\frac{19}{33}$
$\displaystyle a_8=\frac{3(8)-2}{4(8)+5}=\frac{22}{37}$
$\displaystyle \therefore a_7=\frac{19}{33}\text{ and }a_8=\frac{22}{37}.$

$\displaystyle \text{(iii) Given, }a_n=n(n-1)(n-2)$
$\displaystyle a_5=5(5-1)(5-2)=5\times4\times3=60$
$\displaystyle a_8=8(8-1)(8-2)=8\times7\times6=336$
$\displaystyle \therefore a_5=60\text{ and }a_8=336.$

$\displaystyle \text{(iv) Given, }a_n=(-1)^n n$
$\displaystyle a_3=(-1)^3(3)=-3$
$\displaystyle a_5=(-1)^5(5)=-5$
$\displaystyle a_8=(-1)^8(8)=8$
$\displaystyle \therefore a_3=-3,\quad a_5=-5\text{ and }a_8=8.$
$\displaystyle \\$

$\displaystyle \textbf{Question 3: }\text{Find the next five terms of each of the following sequences given by:}$
$\displaystyle \text{(i) }a_1=1,\ a_n=a_{n-1}+2,\ n\geq2$
$\displaystyle \text{(ii) }a_1=a_2=2,\ a_n=a_{n-1}-3,\ n>2$
$\displaystyle \text{Answer:}$
$\displaystyle \text{(i) Given, }a_1=1,\quad a_n=a_{n-1}+2$
$\displaystyle a_2=a_1+2=1+2=3$
$\displaystyle a_3=a_2+2=3+2=5$
$\displaystyle a_4=a_3+2=5+2=7$
$\displaystyle a_5=a_4+2=7+2=9$
$\displaystyle a_6=a_5+2=9+2=11$
$\displaystyle \therefore \text{The next five terms are }3,5,7,9,11.$

$\displaystyle \text{(ii) Given, }a_1=a_2=2,\quad a_n=a_{n-1}-3$
$\displaystyle a_3=a_2-3=2-3=-1$
$\displaystyle a_4=a_3-3=-1-3=-4$
$\displaystyle a_5=a_4-3=-4-3=-7$
$\displaystyle a_6=a_5-3=-7-3=-10$
$\displaystyle a_7=a_6-3=-10-3=-13$
$\displaystyle \therefore \text{The next five terms are }-1,-4,-7,-10,-13.$
$\displaystyle \\$