Answer:
a) 7 +9(n -1)
b) (n +2)² +6
Explanation:
You want expressions for the n-th terms of sequences that start ...
a) 7, 16, 25, 34, 43, ...
b) 15, 22, 31, ...
a)
It is useful to look at the differences of terms of a sequence. Here the "first" differences are ...
16 -7 = 9
25 -16 = 9
34 -25 = 9
43 -34 = 9
The constant differences mean this is an arithmetic sequence. Its first term is a1 = 7, and the common difference is d = 9. The general term is ...
an = a1 +d(n -1)
an = 7 +9(n -1)
b)
The first differences of the numbers of squares are ...
22 -15 = 7
31 -22 = 9
There aren't enough numbers to establish a definitive pattern, but we can assume that the second difference will continue to be 9 -7 = 2. The constant second differences mean the sequence can be described by a quadratic (degree 2) expression.
When the pattern is presented as a geometry, it is often useful to examine the geometry for clues. Here, we see the central body of squares forms a square that has n+2 squares on a side. Those not in that central square total 6, remaining constant from one figure to the next.
The expression for the number of squares can be ...
(n +2)² +6