Question

Find an explicit formula for the following sequence, an, which starts with a1=−4.
−4,1,6,11,16,…

Answer 1

Answer:

$a_{n} = 5n-9$

Step-by-step explanation:

we have  the following sequence,

$-4,1,6,11,16,...$

where

$a_1=-4\\a_2=1\\a_3=6\\a_4=11\\a_5=16$

so

$a_2-a_1=1-(-4)=5$ -----> $a_2=a_1+5$

$a_3-a_2=6-1=5$ -----> $a_3=a_2+5$

$a_4-a_3=11-6=5$ -----> $a_4=a_3+5$

$a_5-a_4=16-116=5$ -----> $a_5=a_4+5$

Note there is a common difference d=5 between consecutive terms

This indicates the sequence is arithmetic with n th term

so

$a_{n} = a₁ + (n - 1)d$

where a₁ is the first term and d the common difference

substitute

$a_{n} = -4 + (n - 1)5$

$a_{n} = -4 +5n-5$

$a_{n} = 5n-9$