Chapter 12, Problem 3RCC

a.

To determine

Explain the meaning of the partial sum of a sequence.

Answer to Problem 3RCC

A partial sum is the sum of part of a sequence.

Explanation of Solution

Given information:

It is given in the question that, to explain the meaning of the partial sum of a sequence.

A partial sum is the sum of part of a sequence.

b.

To determine

Write an expression for the sum of its first $n$ term of arithmetic sequence having first term $a$ and common difference $d$

Answer to Problem 3RCC

The sum of its first $n$ term of arithmetic sequence is

  $S_n=\frac{n}{2}\left[2a+\left(n-1\right)d\right]$

Explanation of Solution

Given information:

It is given in the question that, an arithmetic sequence having first term $a$ and common difference $d$ .

The sum of its first $n$ term of arithmetic sequence is

  $S_n=\frac{n}{2}\left[2a+\left(n-1\right)d\right]$

c.

To determine

Write an expression for the sum of its first $n$ term of geometric sequence having first term $a$ and common ratio $r$

Answer to Problem 3RCC

The sum of its first $n$ term of geometric sequence is

  $S_n=a\left(\frac{1-r^n}{1-r}\right)$

Explanation of Solution

Given information:

It is given in the question that, geometric sequence having first term $a$ and common ratio $d$ .

The sum of its first $n$ term of geometric sequence is

  $S_n=a\left(\frac{1-r^n}{1-r}\right)$ .

d.

To determine

Find an expression for the sum of an infinite geometric series with first term $a$ and common ratio r and also find the value of $r$ is for which formula valid.

Answer to Problem 3RCC

The sum of an infinite geometric sequence is

  $S_n=\frac{a}{1-r}$

And, it is valid for all $r\notin 1$

Explanation of Solution

Given information:

It is given in the question that, infinite geometric series with first term $a$ and common ratio r.

The sum of an infinite geometric sequence is

  $S_n=\frac{a}{1-r}$

And, it is valid for all $r\notin 1$