$3,?,20,39,65,98,138,185$

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Grade 8

Sequences: mixed review

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3,?,20,39,65,98,138,185

Determine Pattern: We need to determine the pattern in the sequence to find the missing number. Let's look at the differences between consecutive terms.
Calculate Differences: First, calculate the difference between the second and third terms: $20 - 3 = 17$ .
Find Missing Number: Now, let's calculate the difference between the third and fourth terms: $39 - 20 = 19$ .
Find Missing Number: Now, let's calculate the difference between the third and fourth terms: $39 - 20 = 19$ .Next, calculate the difference between the fourth and fifth terms: $65 - 39 = 26$ .
Find Missing Number: Now, let's calculate the difference between the third and fourth terms: $39 - 20 = 19$ .Next, calculate the difference between the fourth and fifth terms: $65 - 39 = 26$ .Then, calculate the difference between the fifth and sixth terms: $98 - 65 = 33$ .
Find Missing Number: Now, let's calculate the difference between the third and fourth terms: $39 - 20 = 19$ .Next, calculate the difference between the fourth and fifth terms: $65 - 39 = 26$ .Then, calculate the difference between the fifth and sixth terms: $98 - 65 = 33$ .Now, calculate the difference between the sixth and seventh terms: $138 - 98 = 40$ .
Find Missing Number: Now, let's calculate the difference between the third and fourth terms: $39 - 20 = 19$ .Next, calculate the difference between the fourth and fifth terms: $65 - 39 = 26$ .Then, calculate the difference between the fifth and sixth terms: $98 - 65 = 33$ .Now, calculate the difference between the sixth and seventh terms: $138 - 98 = 40$ .Finally, calculate the difference between the seventh and eighth terms: $185 - 138 = 47$ .
Find Missing Number: Now, let's calculate the difference between the third and fourth terms: $39 - 20 = 19$ .Next, calculate the difference between the fourth and fifth terms: $65 - 39 = 26$ .Then, calculate the difference between the fifth and sixth terms: $98 - 65 = 33$ .Now, calculate the difference between the sixth and seventh terms: $138 - 98 = 40$ .Finally, calculate the difference between the seventh and eighth terms: $185 - 138 = 47$ .The differences between consecutive terms are $17$ , $19$ , $26$ , $33$ , $40$ , and $65 - 39 = 26$ $0$ . It seems that the differences are not increasing by a constant amount, so let's look at the differences between the differences.
Find Missing Number: Now, let's calculate the difference between the third and fourth terms: $39 - 20 = 19$ .Next, calculate the difference between the fourth and fifth terms: $65 - 39 = 26$ .Then, calculate the difference between the fifth and sixth terms: $98 - 65 = 33$ .Now, calculate the difference between the sixth and seventh terms: $138 - 98 = 40$ .Finally, calculate the difference between the seventh and eighth terms: $185 - 138 = 47$ .The differences between consecutive terms are $17, 19, 26, 33, 40,$ and $47$ . It seems that the differences are not increasing by a constant amount, so let's look at the differences between the differences.Calculate the difference between the second and first differences: $19 - 17 = 2$ .
Find Missing Number: Now, let's calculate the difference between the third and fourth terms: $39 - 20 = 19$ .Next, calculate the difference between the fourth and fifth terms: $65 - 39 = 26$ .Then, calculate the difference between the fifth and sixth terms: $98 - 65 = 33$ .Now, calculate the difference between the sixth and seventh terms: $138 - 98 = 40$ .Finally, calculate the difference between the seventh and eighth terms: $185 - 138 = 47$ .The differences between consecutive terms are $17, 19, 26, 33, 40,$ and $47$ . It seems that the differences are not increasing by a constant amount, so let's look at the differences between the differences.Calculate the difference between the second and first differences: $19 - 17 = 2$ .Calculate the difference between the third and second differences: $26 - 19 = 7$ .
Find Missing Number: Now, let's calculate the difference between the third and fourth terms: $39 - 20 = 19$ .Next, calculate the difference between the fourth and fifth terms: $65 - 39 = 26$ .Then, calculate the difference between the fifth and sixth terms: $98 - 65 = 33$ .Now, calculate the difference between the sixth and seventh terms: $138 - 98 = 40$ .Finally, calculate the difference between the seventh and eighth terms: $185 - 138 = 47$ .The differences between consecutive terms are $17, 19, 26, 33, 40,$ and $47$ . It seems that the differences are not increasing by a constant amount, so let's look at the differences between the differences.Calculate the difference between the second and first differences: $19 - 17 = 2$ .Calculate the difference between the third and second differences: $26 - 19 = 7$ .Calculate the difference between the fourth and third differences: $33 - 26 = 7$ .
Find Missing Number: Now, let's calculate the difference between the third and fourth terms: $39 - 20 = 19$ .Next, calculate the difference between the fourth and fifth terms: $65 - 39 = 26$ .Then, calculate the difference between the fifth and sixth terms: $98 - 65 = 33$ .Now, calculate the difference between the sixth and seventh terms: $138 - 98 = 40$ .Finally, calculate the difference between the seventh and eighth terms: $185 - 138 = 47$ .The differences between consecutive terms are $17, 19, 26, 33, 40,$ and $47$ . It seems that the differences are not increasing by a constant amount, so let's look at the differences between the differences.Calculate the difference between the second and first differences: $19 - 17 = 2$ .Calculate the difference between the third and second differences: $26 - 19 = 7$ .Calculate the difference between the fourth and third differences: $33 - 26 = 7$ .Calculate the difference between the fifth and fourth differences: $65 - 39 = 26$ $0$ .
Find Missing Number: Now, let's calculate the difference between the third and fourth terms: $39 - 20 = 19$ .Next, calculate the difference between the fourth and fifth terms: $65 - 39 = 26$ .Then, calculate the difference between the fifth and sixth terms: $98 - 65 = 33$ .Now, calculate the difference between the sixth and seventh terms: $138 - 98 = 40$ .Finally, calculate the difference between the seventh and eighth terms: $185 - 138 = 47$ .The differences between consecutive terms are $17, 19, 26, 33, 40,$ and $47$ . It seems that the differences are not increasing by a constant amount, so let's look at the differences between the differences.Calculate the difference between the second and first differences: $19 - 17 = 2$ .Calculate the difference between the third and second differences: $26 - 19 = 7$ .Calculate the difference between the fourth and third differences: $33 - 26 = 7$ .Calculate the difference between the fifth and fourth differences: $65 - 39 = 26$ $0$ .Calculate the difference between the sixth and fifth differences: $65 - 39 = 26$ $1$ .
Find Missing Number: Now, let's calculate the difference between the third and fourth terms: $39 - 20 = 19$ .Next, calculate the difference between the fourth and fifth terms: $65 - 39 = 26$ .Then, calculate the difference between the fifth and sixth terms: $98 - 65 = 33$ .Now, calculate the difference between the sixth and seventh terms: $138 - 98 = 40$ .Finally, calculate the difference between the seventh and eighth terms: $185 - 138 = 47$ .The differences between consecutive terms are $17, 19, 26, 33, 40,$ and $47$ . It seems that the differences are not increasing by a constant amount, so let's look at the differences between the differences.Calculate the difference between the second and first differences: $19 - 17 = 2$ .Calculate the difference between the third and second differences: $26 - 19 = 7$ .Calculate the difference between the fourth and third differences: $33 - 26 = 7$ .Calculate the difference between the fifth and fourth differences: $65 - 39 = 26$ $0$ .Calculate the difference between the sixth and fifth differences: $65 - 39 = 26$ $1$ .It appears that after the initial difference of $65 - 39 = 26$ $2$ , the differences between the differences are consistently $65 - 39 = 26$ $3$ . This suggests that the sequence of differences follows a pattern starting with $65 - 39 = 26$ $4$ and increasing by $65 - 39 = 26$ $2$ , then by $65 - 39 = 26$ $3$ repeatedly.
Find Missing Number: Now, let's calculate the difference between the third and fourth terms: $39 - 20 = 19$ .Next, calculate the difference between the fourth and fifth terms: $65 - 39 = 26$ .Then, calculate the difference between the fifth and sixth terms: $98 - 65 = 33$ .Now, calculate the difference between the sixth and seventh terms: $138 - 98 = 40$ .Finally, calculate the difference between the seventh and eighth terms: $185 - 138 = 47$ .The differences between consecutive terms are $17, 19, 26, 33, 40,$ and $47$ . It seems that the differences are not increasing by a constant amount, so let's look at the differences between the differences.Calculate the difference between the second and first differences: $19 - 17 = 2$ .Calculate the difference between the third and second differences: $26 - 19 = 7$ .Calculate the difference between the fourth and third differences: $33 - 26 = 7$ .Calculate the difference between the fifth and fourth differences: $65 - 39 = 26$ $0$ .Calculate the difference between the sixth and fifth differences: $65 - 39 = 26$ $1$ .It appears that after the initial difference of $65 - 39 = 26$ $2$ , the differences between the differences are consistently $65 - 39 = 26$ $3$ . This suggests that the sequence of differences follows a pattern starting with $65 - 39 = 26$ $4$ and increasing by $65 - 39 = 26$ $2$ , then by $65 - 39 = 26$ $3$ repeatedly.To find the missing number, we need to determine the difference that should be added to the first term ( $65 - 39 = 26$ $7$ ) to get the second term. Since the pattern of differences increases by $65 - 39 = 26$ $3$ after the initial increase of $65 - 39 = 26$ $2$ , the difference we are looking for should be $98 - 65 = 33$ $0$ .
Find Missing Number: Now, let's calculate the difference between the third and fourth terms: $39 - 20 = 19$ .Next, calculate the difference between the fourth and fifth terms: $65 - 39 = 26$ .Then, calculate the difference between the fifth and sixth terms: $98 - 65 = 33$ .Now, calculate the difference between the sixth and seventh terms: $138 - 98 = 40$ .Finally, calculate the difference between the seventh and eighth terms: $185 - 138 = 47$ .The differences between consecutive terms are $17, 19, 26, 33, 40,$ and $47$ . It seems that the differences are not increasing by a constant amount, so let's look at the differences between the differences.Calculate the difference between the second and first differences: $19 - 17 = 2$ .Calculate the difference between the third and second differences: $26 - 19 = 7$ .Calculate the difference between the fourth and third differences: $33 - 26 = 7$ .Calculate the difference between the fifth and fourth differences: $65 - 39 = 26$ $0$ .Calculate the difference between the sixth and fifth differences: $65 - 39 = 26$ $1$ .It appears that after the initial difference of $65 - 39 = 26$ $2$ , the differences between the differences are consistently $65 - 39 = 26$ $3$ . This suggests that the sequence of differences follows a pattern starting with $65 - 39 = 26$ $4$ and increasing by $65 - 39 = 26$ $2$ , then by $65 - 39 = 26$ $3$ repeatedly.To find the missing number, we need to determine the difference that should be added to the first term ( $65 - 39 = 26$ $7$ ) to get the second term. Since the pattern of differences increases by $65 - 39 = 26$ $3$ after the initial increase of $65 - 39 = 26$ $2$ , the difference we are looking for should be $98 - 65 = 33$ $0$ .Now, add this difference to the first term to find the missing number: $98 - 65 = 33$ $1$ .

3,?,20,39,65,98,138,1853,?,20,39,65,98,138,1853,?,20,39,65,98,138,185

$3,?,20,39,65,98,138,185$