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Type the missing number in this sequence: $\newline$ $5$ , $7$ , $\_$ , $17$ , $25$ , $35$ , $47$

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Sequences: mixed review

Full solution

Q. Type the missing number in this sequence:

\newline

5

,

7

,

\_

,

17

,

25

,

35

,

47

Identify Pattern: $5, 7, \_, 17, 25, 35, 47$ $\newline$ Identify the pattern in the sequence. $\newline$ Look at the differences between consecutive terms to see if there is a consistent pattern.
Calculate Differences: Calculate the differences between the known consecutive terms. $\newline$ $7 - 5 = 2$ $\newline$ $17 - 7 = 10$ $\newline$ $25 - 17 = 8$ $\newline$ $35 - 25 = 10$ $\newline$ $47 - 35 = 12$ $\newline$ The differences are not consistent, but they seem to be increasing. Let's look for another pattern.
Look for Pattern: Look for a pattern in the differences of the differences. $\newline$ $10 - 2 = 8$ $\newline$ $8 - 10 = -2$ $\newline$ $10 - 8 = 2$ $\newline$ $12 - 10 = 2$ $\newline$ The differences of the differences are not consistent either. Let's try another approach.
Consider Formula Involving Squares or Primes: Consider that the sequence might be based on a formula involving squares or primes. $\newline$ Notice that the numbers are not perfect squares or related to prime numbers in a simple way. Let's try looking at the sequence as a series of increasing gaps.
Examine for Increasing Gaps: Examine the sequence again for increasing gaps. $\newline$ $5$ to $7$ is an increase of $2$ . $\newline$ $7$ to $17$ is an increase of $10$ , which is $2^2 + 2$ . $\newline$ $17$ to $25$ is an increase of $8$ , which is $7$ $0$ . $\newline$ $25$ to $7$ $2$ is an increase of $10$ , which is $7$ $4$ . $\newline$ $7$ $2$ to $7$ $6$ is an increase of $7$ $7$ , which is $7$ $8$ . $\newline$ It seems that the increases are multiples of $2$ . Let's see if this pattern holds for the missing number.
Determine Increase to $17$ : Determine the increase needed to go from the missing number to $17$ . If the pattern of increasing by multiples of $2$ holds, the increase before $17$ should be $2 \times 3$ , which is $6$ .
Subtract Increase for Missing Number: Subtract the increase from $17$ to find the missing number. $\newline$ $17 - 6 = 11$ $\newline$ The missing number should be $11$ if the pattern of increasing by multiples of $2$ holds true.
Verify Pattern: Verify the pattern with the newly found number. $\newline$ $5$ to $7$ is an increase of $2$ ( $2$ times $1$ ). $\newline$ $7$ to $11$ is an increase of $4$ ( $2$ times $2$ ). $\newline$ $11$ to $7$ $1$ is an increase of $7$ $2$ ( $2$ times $7$ $4$ ). $\newline$ $7$ $1$ to $7$ $6$ is an increase of $7$ $7$ ( $2$ times $4$ ). $\newline$ $7$ $6$ to $2$ $1$ is an increase of $2$ $2$ ( $2$ times $5$ ). $\newline$ $2$ $1$ to $2$ $6$ is an increase of $2$ $7$ ( $2$ times $7$ $2$ ). $\newline$ The pattern holds true, so the missing number is indeed $11$ .

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