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Form a strong inductive conclusion for the sequence of numbers
3,6,11,20,dots by observing the numbers and its pattern. Hence, find the 6th number for the sequence of numbers. SP 3.2.6

Form a strong inductive conclusion for the sequence of numbers $3,6,11,20, \ldots$ by observing the numbers and its pattern. Hence, find the $6$ th number for the sequence of numbers. SP $3$ . $2$ . $6$

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Write variable expressions for geometric sequences

Full solution

Q. Form a strong inductive conclusion for the sequence of numbers

3,6,11,20, \ldots

by observing the numbers and its pattern. Hence, find the

6

th number for the sequence of numbers. SP

3

.

2

.

6

Observe Pattern: We need to observe the pattern in the sequence to form an inductive conclusion. Let's look at the differences between consecutive terms: $\newline$ $6 - 3 = 3$ $\newline$ $11 - 6 = 5$ $\newline$ $20 - 11 = 9$ $\newline$ It seems that the differences themselves are increasing. To confirm the pattern, let's find the differences between these differences: $\newline$ $5 - 3 = 2$ $\newline$ $9 - 5 = 4$ $\newline$ The second set of differences are increasing by $2$ each time. This suggests that the sequence is formed by adding an increasing odd number to the previous term.
Find $5$ th Term: Let's continue the pattern to find the $5$ th term. The last difference we had was $9$ , so the next difference should be $9 + 2 = 11$ . Now, we add this difference to the $4$ th term to find the $5$ th term: $\newline$ $20 + 11 = 31$ $\newline$ So, the $5$ th term is $31$ .
Find $6$ th Term: Now, we need to find the $6^{th}$ term. The next difference will be $11 + 2 = 13$ . We add this difference to the $5^{th}$ term to find the $6^{th}$ term: $\newline$ $31 + 13 = 44$ $\newline$ So, the $6^{th}$ term is $44$ .

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