Question

(a) Write the algebraic form of the arithmetic sequence $1,4,7,10,..$
(b) Is $100$ a term of this sequence? Why?
(c) Prove that the square of any term of this sequence is also a term of it.

Answer 1

Sequence is $1,4,7,10,.....$
(a) $a=1,d=4-1=3$
$t_n=a+(n-1)d$
$\Rightarrow t_n=1+(n-1)d$
$\Rightarrow t_n=1+3n-3$
$\Rightarrow t_n=3n-2$ which is algebraic form of the given arithmetic sequence.
(b) $t_n=3n-2$
$\Rightarrow t_n=3n-2$
$\Rightarrow 3n=102$
$\Rightarrow n=34$
$\Rightarrow 100$ is the ${34}^{th}$ term of the sequence.
(c) We know, $t_n=3n-2$
Square of the term of this sequence
$=(3n-2{)}^2$
$=9n^2-12n+4$
$=9n^2-12n+12-8$
$=3[3n-2{)}^2-2]-2$
Let $k=\left[\right(3n-2{)}^2-2]$
Square of the term of this sequence $=3k-2$
Since square of the sequence is of the form $=3k-2$, therefore square of any term of the sequence is a term of this sequence.