Question

1. an arithmetic sequence has a 7th term of 54 and a 13th term of 94. find the common difference.

Explanation:

Step1: Recall arithmetic sequence formula

The nth term of an arithmetic sequence is given by $a_n = a_1 + (n-1)d$, where $a_1$ is the first term, $d$ is the common difference, and $n$ is the term number.

Step2: Set up equations for given terms

For the 7th term: $54 = a_1 + (7-1)d \implies 54 = a_1 + 6d$
For the 13th term: $94 = a_1 + (13-1)d \implies 94 = a_1 + 12d$

Step3: Subtract equations to eliminate $a_1$

Subtract the first equation from the second:
$94 - 54 = (a_1 + 12d) - (a_1 + 6d)$
$40 = 6d$

Step4: Solve for $d$

$d = \frac{40}{6} = \frac{20}{3} \approx 6.67$

Answer:

The common difference is $\frac{20}{3}$ (or approximately 6.67)