Find the first term or a1 of an arithmetic sequence if a8 = 64 and a23 = 184


#1

Find the first term or a1 of an arithmetic sequence if
a8 = 64 and a23 = 184.

Answer:

Let ‘d’ be the common difference.

Now Expression for nth term of Arithmetic series is
an=a1+(n-1)d;----------------->(1)

Given: a8=64 and a23=184
=>a1+(8-1)d=64 ------->(2) & a1+(23-1)d=184------->(3)

(3)-(2)=>22d-7d=184-64=120=>15d=120=>d=8

put d=8 in (1)=>a1+(7)(8)=64=>a1=64-56=8

Okay so you’re giving an arithmetic sequence which increases in value as it progresses. So we know that it’s a positive number being added each time.

There are two formulas you can use for an arithmetic sequence (if you are studying about sequences I suggest you memorize or know these very well):

an=a1+(n−1)×d

OR

an=am+(n−m)×d

where:
d : is the common difference between each term
n & m : the number of term in the sequence, for example… a1 means first term

We do not know the common difference, d, yet and we an find that out using simple math:

184 - 64 = 120 <— this is the number DIFFERENCE between the entire two given values

23 - 8 - 15 <---- this is the number of INTERVALS between the two values

120 / 15 = 8 <---- this is the COMMON DIFFERENCE between each interval

So using the formula we get 64 = a1 + (8-1) x 8
Remember 64 is a(n) and n=8 because it was the 8th term

a1 = 8