Question

1. in the sequence 65, 72, 79, 86,.... (3pts)

a) what is the first term?
the first term is 65,72,79,86... is 65
b) what is the common difference?
c) what is the 16th term in the sequence?

Explanation:

Part a)

The first term of a sequence is the initial number in the sequence. Looking at 65, 72, 79, 86,..., the first term is the first number listed.

Step1: Recall common difference formula

For an arithmetic sequence, the common difference \( d = a_{n + 1}-a_{n}\), where \( a_{n+1}\) is the next term and \( a_{n}\) is the current term.

Step2: Calculate using given terms

Take \( a_{1}=65\) and \( a_{2}=72\). Then \( d = 72 - 65=7\). We can verify with other terms: \( 79 - 72 = 7\), \( 86 - 79 = 7\), so the common difference is consistent.

Step1: Recall arithmetic sequence formula

The formula for the \( n\)-th term of an arithmetic sequence is \( a_{n}=a_{1}+(n - 1)d\), where \( a_{1}\) is the first term, \( d\) is the common difference, and \( n\) is the term number.

Step2: Substitute values

We know \( a_{1}=65\), \( d = 7\), and \( n = 16\). Substitute into the formula: \( a_{16}=65+(16 - 1)\times7\).

Step3: Simplify the expression

First, calculate \( 16 - 1 = 15\). Then, \( 15\times7 = 105\). Finally, \( 65+105 = 170\).

Answer:

The first term is 65.

Part b)