number patterns

wha would be the next 2 numbers in the pattern 1,5,13,25

1. Calculate the differences. You'll get an arithmetic sequence.

2. Augment the sequence of differences by 2 elements and use them to determine the following 2 elements of the given sequence.
 
Because, as pappus says, the first differences are an arithmetic sequence, the second differences are constant. One thing that tells us is that the sequential numbers are a quadratic function of n: Write y(n)=an2+bn+c\displaystyle y(n)= an^2+ bn+ c. Knowing that y(0)= c= 1, y(1)= a+ b+ c= 5, and y(2)= 4a+ 2b+ c= 13, you can solve for a, b, and c. Using those values, find y(3) as a check (it should be equal to 25), then find y(4) and y(5).
 
Hello, nolikemath!

What would be the next 2 numbers in the pattern? .1, 5, 13, 25, . . .

Look at the "gaps" between the numbers . . .

. . Sequence:151325Gaps:4812\displaystyle \begin{array}{ccccccccccc} \text{Sequence:} &1 && 5 && 13 && 25 & \cdots \\ \hline \text{Gaps:} && 4 && 8 && 12 & \cdots\end{array}

The gaps are increasing by 4 each time.
The next two gaps are 16 and 20.

The next two terms are: .{25+16=4141+20=61}\displaystyle \begin{Bmatrix} 25 + 16 &=& 41 \\ 41 + 20 &=& 61 \end{Bmatrix}
 
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