Question 11.11.2.5: Finding a Recursive formula for an arithmetic Sequence The 8......
Finding a Recursive formula for an arithmetic
Sequence The 8th term of an arithmetic sequence is 75, and the 20th term is 39.
(a) Find the first term and the common difference.
(b) Give a recursive formula for the sequence.
(c) What is the nth term of the sequence?
(a) Formula (2) states that a_{n}=a_{1}+\;(n\;-\;1)d. As a result,
\begin{cases} a_{8} = a_{1} + 7d = 75 \\ a_{20} = a_{1} + 19d = 39 \end{cases}This is a system of two linear equations containing two variables, a_{1} and d, which can be solved by elimination. Subtracting the second equation from the first gives
\begin{array}{c}{{-12d=36}}\\ {{d=-3}}\end{array}With d = -3, use a_{1}+7d=75 to find that a_{1}=75-7d=75-7(-3)= 96. The first term is a_{1}=96, and the common difference is d = -3.
(b) Using formula (1), a recursive formula for this sequence is
a_{1}=a,\qquad a_{n}=a_{n-1}+d (1)
a_{1}=96,\qquad a_{n}=a_{n-1}-3(c) Using formula (2), a formula for the nth term of the sequence
\{\,a_{n}\} is a_{n}=a_{1}+\;(n\;-\;1)d=96+\;(n\;-\;1)\,(-3)\;=\;99\,-\;3n