Question

Shaunta is developing a recursive formula to represent an arithmetic sequence in which 5 is added to each term to determine each successive term. Which formula could represent her sequence?

Answer 1

Given:-

An arithmetic sequence in which 5 is added to each term to determine each successive term, i.e., 5 is added to each term to determine successive terms.

⇒$d=5$

To Find:

We have to find the formula that could represent her sequence.

Solution:-

• Shaunta can use Arithmetic progression to represent her sequence.

• Let us assume the first term to be '$a$'.

• ' $a$' can have any value which Shaunta desires. It will be the first term of her sequence. It can have any value let it be a fraction, decimal, whole number, integer, or even irrational values.

• For the next term of the sequence, $5$ should be added to '$a$'.

        ⇒ $a$₂$= a+d$   will be the second term of the sequence.

       Similarly for the third term $a$₃ we need to add 5 to

       ⇒ $a$₃$= a+d+d = a+2d$

• If you notice carefully for every successive term we need to add '$d$ ' one times less than the number of that term, for example for the second term '$a$₂'.

• Therefore, if a sequence has $n$ terms, you need to add $d$ for $(n-1)$ times.

$\therefore$ The general formula for the arithmetic progression becomes

$a$ₙ$=a+(n-1)d$

and the specific formula representing Shaunta's sequence becomes

$a$ₙ$=a+(n-1)5$