Find a formula for a_n for the arithmetic sequence. a_3=19, a_13=99?

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Answer 1 · 2018-04-16T17:54:01

#color(blue)(a_n=8n-5)#

The nth term of an arithmetic sequence is given as:

#a+(n-1)d#

Where #a# is the first term, #d# is the common difference and #n# is the nth term.

We are given:

#a_3=19#

#a_13=99#

Using the above for the nth term:

#a+(3-1)d=19#

#a+(13-1)d=99#

#a+2d=19 \ \ \ \ \ \ \ \ \ \ \ [1]#

#a+12d=99\ \ \ \ \ \ \ \[2]#

Solving for #a and d# simultaneously:

From #[1]#

#a=19-2d#

Substituting in#[2]#

#19-2d+12d=99#

#10d=80#

#d=80/10=8#

Substituting this is #[1]#

#a+2(8)=19#

#a+16=19#

#a=3#

#:.#

#a_n=3+(n-1)8=3+8n-8=color(blue)(8n-5)#