To find the first term of an arithmetic sequence when you have two terms, let’s denote the two known terms as a and b, where a is the earlier term and b is the later term. The formula for the nth term of an arithmetic sequence is given by:
Tn = a1 + (n – 1)d
Here, a1 is the first term, and d is the common difference between consecutive terms.
First, we need to find the common difference d between the two terms:
d = b – a
Next, let’s identify the positions of the terms a and b. Suppose a is the mth term and b is the nth term. We can express a and b in terms of a1 and d:
a = a1 + (m – 1)d
b = a1 + (n – 1)d
To find the first term a1, rearrange the equation for a:
a1 = a – (m – 1)d
Now substitute d from the first formula:
a1 = a – (m – 1)(b – a)
After calculating this, you’ll arrive at the first term of the arithmetic sequence. This process allows you to derive the first term from any two known terms in a sequence.