Number - Number sequences (prediction, description)

Find the $n^{th}$ term formula for the arithmetic sequence: $7, 11, 15, 19, \dots$

$u_n = 4n + 3$

First, find the common difference $d$: $11 - 7 = 4$. The formula follows $u_n = dn + c$, so $u_n = 4n + c$. To find $c$, use the first term ($n=1$): $4(1) + c = 7$, which gives $c = 3$. Therefore, $u_n = 4n + 3$.

Determine the $10^{th}$ term of the geometric sequence: $3, 6, 12, 24, \dots$

$u_{10} = 1536$

Identify the first term $a = 3$ and the common ratio $r = \frac{6}{3} = 2$. Use the formula $u_n = ar^{n-1}$. For the $10^{th}$ term: $u_{10} = 3 \times 2^{10-1} = 3 \times 2^9 = 3 \times 512 = 1536$.

Find the $n^{th}$ term of the sequence: $4, 9, 16, 25, \dots$

$u_n = (n+1)^2$

Observe that these are square numbers. The standard square numbers are $1^2, 2^2, 3^2, \dots$. Comparing $1, 4, 9, 16$ with $4, 9, 16, 25$, we see the sequence is shifted. When $n=1, u_1 = (1+1)^2 = 4$. When $n=2, u_2 = (2+1)^2 = 9$. Thus, $u_n = (n+1)^2$.

Find the next two terms in the sequence: $1, 3, 6, 10, \dots$

$15, 21$

This is the sequence of Triangle Numbers. The difference between terms increases by 1 each time: $3-1=2$, $6-3=3$, $10-6=4$. The next difference will be $5$ ($10+5=15$) and the one after will be $6$ ($15+6=21$).