Given the nth term for each arithmetic sequence, how to find the common difference and write out the first four terms here? (1) #a_n=2n+7# (2) #a_n=3n-2#

1 Answer
KillerBunny
Jun 27, 2018

See below.

Explanation:

Since every couple of consecutive terms in an arithmetic sequence differ by a common difference, we can subtract any two consecutive terms to find out how distant they are from each other.

So, in the first case, let's consider the #n^"th"# and the #n+1^"th"# term:

#a_n=2n+7,\qquad a_{n+1}=2(n+1)+7 = 2n+2+7=2n+9#

and subtract them:

#a_{n+1}-a_n=2n+9-(2n+7)=cancel(2n)+9-cancel(2n)-7=2#

As for the first four terms, it depends if you consider the first term to be associated with #n=0# or #n=1#. I usually go for the first choice. With this assumption, the first four terms are

#a_0 = 2*0 + 7 = 0+7 = 7#
#a_1 = 2*1 + 7 = 2+7 = 9#
#a_2 = 2*2 + 7 = 4+7 = 11#
#a_3 = 2*3 + 7 = 6+7 = 13#

In other words, you simply have to plug the value of #n# in the expression for the general term.

The second sequence behaves exactly in the same way.

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