Question

PLS HELP!! AND HURRY!! Find an expression for the nth term ( tn ) in each of the following sequences:

Answer 1

If the pattern continues, so that each term is separated by a distance of 3, then the sequence is given by the recursive rule

$\begin{cases}a_1=1\\a_n=a_{n-1}+3&\text{for }n>1\end{cases}$

From this definition, we can write $a_n$ in terms of $a_1$:

$a_2=a_1+3$

$a_3=a_2+3=(a_1+3)+3=a_1+2\cdot3$

$a_4=a_3+3=(a_1+2\cdot3)+3=a_1+3\cdot3$

$a_5=a_4+3=(a_1+3\cdot3)+3=a_1+4\cdot3$

and so on, up to

$a_n=a_1+(n-1)\cdot3$

(notice how the subscript on a and coefficient on 3 add up to n)

or

$a_n=1+3(n-1)=3n-2$