Question

find a formula for the nth term in this arithmetic sequence:
a₁ = 28, a₂ = 21, a₃ = 14, a₄ = 7, ...
aₙ = ?n +
hint: the coefficient of n is the common difference.

Explanation:

Step1: Find the common difference (d)

The common difference \( d \) in an arithmetic sequence is \( a_{n + 1}-a_{n} \). Let's calculate \( d \) using \( a_2 - a_1 \):
\( d=a_2 - a_1=21 - 28=- 7 \)

Step2: Recall the formula for the nth term of an arithmetic sequence

The formula for the nth term of an arithmetic sequence is \( a_{n}=a_{1}+(n - 1)d \). We can also rewrite it in the form \( a_{n}=dn+(a_{1}-d) \) (by expanding \( a_{1}+(n - 1)d=a_{1}+dn - d=dn+(a_{1}-d) \)).
We know \( a_1 = 28 \) and \( d=-7 \). Let's find the constant term:
\( a_{1}-d=28-(-7)=28 + 7=35 \)
So the formula for the nth term is \( a_{n}=-7n + 35 \)

Answer:

\( a_{n}=-7n + 35 \) (So the coefficient of \( n \) is \(-7\) and the constant term is \( 35 \))