How do you write the nth term rule for the arithmetic sequence with a_7=34 and a_18=122?

2 Answers
Jul 31, 2016

n^(th) term of the arithmetic sequence is 8n-22.

Explanation:

n^(th) term of an arithmetic sequence whose first term is a_1 and common difference is d is a_1+(n-1)d.

Hence a_7=a_1+(7-1)xxd=34 i.e. a_1+6d=34

and a_18=a_1+(18-1)xxd=122 i.e. a_1+17d=122

Subtracting firt equation from second equation, we get

11d=122-34=88 or d=88/11=8

Hence a_1+6xx8=34 or a_1=34-48=-14

Hence n^(th) term of the arithmetic sequence is -14+(n-1)xx8 or -14+8n-8=8n-22.

color(blue)(a_n=8n-22)

Explanation:

The given data are

a_7=34 and a_18=122

We can set up 2 equations

a_n=a_1+(n-1)*d

a_7=a_1+(7-1)*d

34=a_1+6*d" "first equation
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
a_n=a_1+(n-1)*d

a_18=a_1+(18-1)*d

122=a_1+17*d" "second equation

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

By method of elimination using subtraction, let us use first and second equations

34=a_1+6*d" "first equation

122=a_1+17*d" "second equation

By subtraction, we have the result

88=0+11d

d=88/11=8

Solving now for a_1 using the first equation and d=8

34=a_1+6*d" "first equation

34=a_1+6*8" "

34=a_1+48

a_1=-14

We can write the nth term rule now

#a_n=-14+8*(n-1)

a_n=-14-8+8n

color(blue)(a_n=8n-22)

God bless....I hope the explanation is useful.