Question

Write a recursive formula and an explicit formula for the following arithmetic sequence.

Answer 1

Answer:

Please check the explanation.

Step-by-step explanation:

Given the sequence

11, 13, 15, 17, 19, ...

Determining the Recursive formula:

We know that a recursive formula is termed as a formula that specifies each term of the given sequence using the preceding terms.

From the given sequence it is clear that every term can be obtained by adding two to the previous term.

i.e. 13 = 11+2, 15 = 13+2, 17 = 15+2, 19 = 17+2

so

aₙ₊₁ = aₙ+2, for n ≥1

Therefore, a recursive formula is:

• aₙ₊₁ = aₙ+2, for n ≥1

Determining the Explicit formula:

Given the sequence

11, 13, 15, 17, 19, ...

An arithmetic sequence has a constant difference 'd' and is defined by  

$a_n=a_1+\left(n-1\right)d$

computing the differences of all the adjacent terms

$3-11=2,\:\quad \:15-13=2,\:\quad \:17-15=2,\:\quad \:19-17=2$

The difference between all the adjacent terms is the same and equal to

$d=2$

also

$a_1=11$

so substituting $d=2$, $a_1=11$ in the nth terms

$a_n=a_1+\left(n-1\right)d$

$a_n=2\left(n-1\right)+11$

$a_n=2n+9$

Therefore, the Explicit formula is:

$a_n=2n+9$