How do you find a general formula for each arithmetic sequence given 8th term is -20; 17th term is -47?

Question

Liam Brennan · Accepted Answer

#n^(th)# term of the arithmetic sequence is given by #4-3n#

If #a# is the first term of an arithmetic sequence and #d# the difference between a term and its preceding term, general formula for #n^(th)# term of the arithmetic sequence is given by #a+(n-1)d#.

As #8^(th)# term is #-20# and #17^th# term is #-47#

#a+(8-1)d=a+7d=-20# and #a+(17-1)d=a+16d=-47#.

Subtracting first equation from second, we get

#9d=-47+20# or #9d=-27# i.e. #d=-3#

putting this in first we get #a+7*(-3)=-20# or #a=1#.

Hence, #n^(th)# term of the arithmetic sequence is given by #1-3(n-1)# or #4-3n#

Elias Ackerman · Answer

undefined To find the general formula for an arithmetic sequence, you can use the formula for the $n$th term of an arithmetic sequence:

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

Where:
- $ a_n $ is the $n$th term of the sequence.
- $ a_1 $ is the first term of the sequence.
- $ d $ is the common difference between consecutive terms.
- $ n $ is the term number.

Given:
The 8th term ($ a_8 $) is -20.
The 17th term ($ a_{17} $) is -47.

1) Using the formula for the 8th term:
$$ a_8 = a_1 + 7d = -20 $$
$$ a_1 + 7d = -20 $$ ... (i)

2) Using the formula for the 17th term:
$$ a_{17} = a_1 + 16d = -47 $$
$$ a_1 + 16d = -47 $$ ... (ii)

To find $ a_1 $ and $ d $, solve the system of equations (i) and (ii).

Subtracting (i) from (ii) to eliminate $ a_1 $:
$$ 16d - 7d = -47 + 20 $$
$$ 9d = -27 $$
$$ d = -3 $$

Substituting $ d = -3 $ into (i) to find $ a_1 $:
$$ a_1 + 7(-3) = -20 $$
$$ a_1 - 21 = -20 $$
$$ a_1 = 1 $$

The general formula for the arithmetic sequence is:
$$ a_n = 1 + (n - 1)(-3) $$
$$ a_n = 1 - 3n + 3 $$
$$ a_n = 4 - 3n $$