Question

82.) find the first term of a sequence whose 205th term is -611 and whose common difference is -6.

Explanation:

Step1: Recall the arithmetic sequence formula

The formula for the \(n\)-th term of an arithmetic sequence 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.

Step2: Substitute the given values into the formula

We know that \(a_{205}=-611\), \(n = 205\), and \(d=-6\). Substituting these into the formula \(a_n=a_1+(n - 1)d\), we get:
\(-611=a_1+(205 - 1)\times(-6)\)

Step3: Simplify the equation

First, calculate \((205 - 1)\times(-6)\): \(204\times(-6)=-1224\)
So the equation becomes \(-611=a_1-1224\)

Step4: Solve for \(a_1\)

To solve for \(a_1\), add 1224 to both sides of the equation:
\(a_1=-611 + 1224\)
\(a_1 = 613\)

Answer:

The first term of the sequence is \(\boldsymbol{613}\).