Question

the first term of an arithmetic progression is -3. if the sixth term is 22, find the mean of the common difference and the first term.
a. 1
b. 4
c. 5
d. 7
e. 8

Explanation:

Step1: Recall the formula for the nth term of an arithmetic progression

The formula for the \(n\)th term of an arithmetic progression is \(a_n=a_1+(n - 1)d\), where \(a_n\) is the \(n\)th term, \(a_1\) is the first term, \(n\) is the term number, and \(d\) is the common difference. Given \(a_1=-3\) and \(n = 6\), \(a_6=a_1+(6 - 1)d\).

Step2: Substitute the values into the formula

We know \(a_6 = 22\) and \(a_1=-3\). Substituting into \(a_n=a_1+(n - 1)d\) gives \(22=-3+(6 - 1)d\).

Step3: Solve for \(d\)

First, simplify the equation: \(22=-3 + 5d\). Then, add \(3\) to both sides: \(22+3=5d\), so \(25 = 5d\). Divide both sides by \(5\): \(d=\frac{25}{5}=5\).

Step4: Find the difference \(d - a_1\)

Substitute \(d = 5\) and \(a_1=-3\) into \(d - a_1\). We get \(5-(-3)=5 + 3=8\).

Answer:

E. 8