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Math
Question 7

What number comes next in this sequence?

" 1,4,8,13, 19, "26

$2$ . Math $\newline$ Question $7$ $\newline$ What number comes next in this sequence? $\newline$ $\text { 1,4,8,13, 19, } 26$

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Sequences: mixed review

Full solution

Q.

2

. Math

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Question

7

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What number comes next in this sequence?

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\text { 1,4,8,13, 19, } 26

Identify Differences in Sequence: To find the pattern in the sequence, we need to look at the differences between consecutive numbers. $\newline$ $1$ to $4$ is an increase of $3$ . $\newline$ $4$ to $8$ is an increase of $4$ . $\newline$ $8$ to $13$ is an increase of $5$ . $\newline$ $13$ to $4$ $0$ is an increase of $4$ $1$ . $\newline$ $4$ $0$ to $4$ $3$ is an increase of $4$ $4$ . $\newline$ It seems that the pattern involves increasing the difference by $1$ each time.
Predict Next Number: To predict the next number, we should add $8$ to the last number in the sequence, since the last increase was by $7$ . $\newline$ $26 + 8 = 34$
Verify Pattern Continuity: We should check the pattern to ensure that the difference is indeed increasing by $1$ each time. $\newline$ $1$ to $4$ : $+3$ $\newline$ $4$ to $8$ : $+4$ $\newline$ $8$ to $13$ : $+5$ $\newline$ $13$ to $1$ $1$ : $1$ $2$ $\newline$ $1$ $1$ to $1$ $4$ : $1$ $5$ $\newline$ The pattern holds, so the next difference should be $1$ $6$ .

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Question

Does the infinite geometric series converge or diverge?

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1 + \frac{3}{4} + \frac{9}{16} + \frac{27}{64} + \cdots

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