Resources

Resources

Bytelearn - cat image with glasses

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

Type the missing number in this sequence: $\newline$ $5$ , $7$ , $11$ , $\_$ , $25$ , $35$ , $47$ , $61$

View step-by-step help

View step-by-step help

Sequences: mixed review

Full solution

Q. Type the missing number in this sequence:

\newline

5

,

7

,

11

,

\_

,

25

,

35

,

47

,

61

Identify Pattern: $5, 7, 11, \_\_\_\_, 25, 35, 47, 61$ $\newline$ Identify the pattern in the sequence. $\newline$ We need to observe the differences between consecutive terms to see if there is a consistent pattern.
Calculate Differences: Calculate the differences between consecutive terms that are given. $\newline$ $7 - 5 = 2$ $\newline$ $11 - 7 = 4$ $\newline$ The differences are not consistent so far, so let's look at the next set of differences. $\newline$ $25 - 11 = 14$ $\newline$ $35 - 25 = 10$ $\newline$ $47 - 35 = 12$ $\newline$ $61 - 47 = 14$ $\newline$ The differences are not consistent, but they are increasing. We need to find the pattern in the differences.
Look for Patterns: Look for a pattern in the differences of the differences. $\newline$ $4 - 2 = 2$ $\newline$ $14 - 4 = 10$ $\newline$ $10 - 14 = -4$ $\newline$ $12 - 10 = 2$ $\newline$ $14 - 12 = 2$ $\newline$ It seems that the differences of the differences are not consistent either. We need to look for another pattern.
Analyze Numbers: Analyze the sequence of given numbers to identify if they are special numbers, such as prime numbers. $\newline$ $5$ is a prime number. $\newline$ $7$ is a prime number. $\newline$ $11$ is a prime number. $\newline$ $25$ is not a prime number, but it is a square number $(5^2)$ . $\newline$ $35$ is not a prime number $(5 \times 7)$ . $\newline$ $47$ is a prime number. $\newline$ $61$ is a prime number. $\newline$ It seems that most of the numbers are prime numbers, except for $25$ and $35$ . Let's see if the missing number is a prime number.
Check Prime Numbers: Check the prime numbers between $11$ and $25$ to find the missing number. $\newline$ The prime numbers between $11$ and $25$ are $13$ , $17$ , $19$ , and $23$ . $\newline$ Since $25$ is not a prime number, we can assume that the sequence is not strictly prime numbers. However, $25$ is the square of $25$ $0$ , and $25$ $1$ is the product of $25$ $0$ and $25$ $3$ , which are both prime. Let's see if the missing number is related to prime numbers in a similar way.
Analyze Relationships: Analyze the relationship between the non-prime numbers and the prime numbers in the sequence. $\newline$ $25$ is $5$ squared, and $5$ is the prime number before $7$ . $\newline$ $35$ is the product of $5$ and $7$ , which are consecutive prime numbers. $\newline$ The missing number should be related to the prime numbers before and after $11$ , which are $7$ and $13$ .
Calculate Missing Number: Calculate the missing number based on the relationship between the non-prime numbers and the prime numbers. $\newline$ The missing number should be the product of $7$ and $13$ , as $35$ is the product of $5$ and $7$ . $\newline$ $7 \times 13 = 91$
Verify Solution: Verify if $91$ fits the pattern in the sequence. $\newline$ $91$ is not a prime number, but it is the product of two consecutive prime numbers ( $7$ and $13$ ), similar to how $35$ is the product of $5$ and $7$ . $\newline$ The sequence seems to be a mix of prime numbers and products of consecutive prime numbers.

More problems from Sequences: mixed review

Question

Find the sum of the finite arithmetic series.

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

\newline

Posted 3 months ago

Question

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

Posted 2 months 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 2 months 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 2 months 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 2 months 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 2 months 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 3 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 2 months ago