Resources

Resources

Bytelearn - cat image with glasses

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

Write an explicit formula that represents the sequence defined by the following recursive formula:

a_(1)=1" and "a_(n)=-4a_(n-1)
Answer:
a_(n)=

Write an explicit formula that represents the sequence defined by the following recursive formula: $\newline$ $a_{1}=1 \text { and } a_{n}=-4 a_{n-1}$ $\newline$ Answer: $a_{n}=$

View step-by-step help

View step-by-step help

Convert an explicit formula to a recursive formula

Full solution

Q. Write an explicit formula that represents the sequence defined by the following recursive formula:

\newline

a_{1}=1 \text { and } a_{n}=-4 a_{n-1}

\newline

Answer:

a_{n}=

Find Terms: To find the explicit formula for the sequence, we need to determine the pattern of the sequence based on the given recursive formula. Let's start by finding the first few terms of the sequence. $\newline$ $a_{1} = 1$ (given) $\newline$ $a_{2} = -4a_{1} = -4(1) = -4$ $\newline$ $a_{3} = -4a_{2} = -4(-4) = 16$ $\newline$ $a_{4} = -4a_{3} = -4(16) = -64$ $\newline$ We can see that the sequence is alternating in sign and each term is $4$ times the previous term in magnitude.
Identify Pattern: Now, let's look for a pattern in the exponents of $4$ for each term. $\newline$ $a_{1} = 4^0$ (since $4^0 = 1$ ) $\newline$ $a_{2} = -4^1$ $\newline$ $a_{3} = 4^2$ $\newline$ $a_{4} = -4^3$ $\newline$ We can see that the exponent of $4$ is one less than the term number and the sign alternates with each term.
Sign Pattern: To account for the alternating sign, we can use $(-1)$ raised to a power that will alternate the sign for each term. We can use the term number minus $1$ as the exponent for $(-1)$ to alternate the sign. $\newline$ So, the pattern for the sign is $(-1)^{(n-1)}$ .
Combine Patterns: Combining the pattern for the magnitude of $4$ and the alternating sign, we can write the explicit formula for the sequence as: $\newline$ $a_{n} = (-1)^{n-1} \times 4^{n-1}$ $\newline$ This formula will give us the $n$ th term of the sequence by plugging in the value of $n$ .

More problems from Convert an explicit formula to a recursive 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