Resources

Resources

Bytelearn - cat image with glasses

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

Use the initial term and the recursive formula to find an explicit formula for the sequence $a_n$ . Write your answer in simplest form. $\newline$ $a_1 = 3$ $\newline$ $a_n = a_{n - 1} + 1$ $\newline$ $a_n =$ ______

View step-by-step help

View step-by-step help

Convert a recursive formula to an explicit formula

Full solution

Q. Use the initial term and the recursive formula to find an explicit formula for the sequence

a_n

. Write your answer in simplest form.

\newline

a_1 = 3

\newline

a_n = a_{n - 1} + 1

\newline

a_n =

______

Given initial term and recursive formula: We are given the initial term of the sequence as $a_1 = 3$ and the recursive formula $a_n = a_{n - 1} + 1$ . We need to determine if the sequence is arithmetic or geometric to find the explicit formula.
Determining if the sequence is arithmetic or geometric: The recursive formula $a_n = a_{n - 1} + 1$ suggests that each term is obtained by adding $1$ to the previous term. This is characteristic of an arithmetic sequence, where the common difference $(d)$ between consecutive terms is constant.
Finding the common difference: To find the common difference $d$ , we compare the recursive formula $a_n = a_{n - 1} + 1$ with the general form of an arithmetic sequence $a_n = a_1 + (n - 1)d$ . It is clear that $d = 1$ .
Using the explicit formula for arithmetic sequence: The explicit formula for an arithmetic sequence is given by $a_n = a_1 + d(n - 1)$ , where $a_1$ is the first term and $d$ is the common difference. We will use this formula to find the explicit formula for our sequence.
Substituting values into the explicit formula: Substitute the given values $a_1 = 3$ and $d = 1$ into the explicit formula $a_n = a_1 + d(n - 1)$ to get $a_n = 3 + 1(n - 1)$ .
Simplifying the expression: Simplify the expression $a_n = 3 + 1(n - 1)$ to get $a_n = 3 + n - 1$ .
Simplifying the expression: Simplify the expression $a_n = 3 + 1(n - 1)$ to get $a_n = 3 + n - 1$ . Further simplification gives us $a_n = n + 2$ . This is the explicit formula for the given sequence.

More problems from Convert a recursive formula to an explicit formula

Question

Find the sum of the finite arithmetic series.

\sum_{n=1}^{10} (7n+4)

\newline

Posted 2 months ago

Question

Type the missing number in this sequence: `55,\59,\63,\text{_____},\71,\75,\79`

Posted 1 month ago

Question

Type the missing number in this sequence: `2, 4 8`, ______,`32`

Posted 2 months ago

Question

What kind of sequence is this?

2, 10, 50, 250, \ldots

Choices:Choices:

\newline

\text{[A]arithmetic}

\newline

\text{[B]geometric}

\newline

\text{[C]both}

\newline

\text{[D]neither}

Posted 2 months ago

Question

What is the missing number in this pattern?

1, 4, 9, 16, 25, 36, 49, 64, 81, \_\_\_\_

Posted 1 month ago

Question

Classify the series.

\sum_{n = 0}^{12} (n + 2)^3

\newline

\newline

\text{[A]arithmetic}

\newline

\text{[B]geometric}

\newline

\text{[C]both}

\newline

\text{[D]neither}

Posted 1 month ago

Question

Find the first three partial sums of the series.

\newline

1 + 6 + 11 + 16 + 21 + 26 + \cdots

\newline

Write your answers as integers or fractions in simplest form.

\newline

S_1 =

\newline

S_2 =

\newline

S_3 =

Posted 1 month ago

Question

Find the third partial sum of the series.

\newline

3 + 9 + 15 + 21 + 27 + 33 + \cdots

\newline

Write your answer as an integer or a fraction in simplest form.

\newline

S_3 =

Posted 1 month ago

Question

Find the first three partial sums of the series.

\newline

1 + 7 + 13 + 19 + 25 + 31 + \cdots

\newline

Write your answers as integers or fractions in simplest form.

\newline

S_1 =

\newline

S_2 =

\newline

S_3 =

Posted 2 months ago

Question

Does the infinite geometric series converge or diverge?

\newline

1 + \frac{3}{4} + \frac{9}{16} + \frac{27}{64} + \cdots

\newline

\newline

\text{[A]converge}

\newline

\text{[B]diverge}

Posted 1 month ago