question 2 predict the nth term in the following patterns. c) 1, 6, 15, 28, ...

Explanation:

Step1: Find differences between terms

The differences between consecutive terms are: $6 - 1=5$, $15 - 6 = 9$, $28 - 15=13$. The differences $5,9,13$ form an arithmetic - sequence with a common difference of $4$.

Step2: Express the $n$th term formula

Let the $n$th term of the sequence be $a_n$. We know that the formula for the $n$th term of a sequence can be found by considering the cumulative sum of the differences. The first - order differences form an arithmetic sequence with first term $d_1 = 5$ and common difference $4$. The sum of the first $(n - 1)$ terms of this arithmetic sequence of differences gives the increment from the first term $a_1$ to the $n$th term $a_n$. The sum of an arithmetic sequence $S_m=\sum_{k = 1}^{m}(a_1+(k - 1)d)=\frac{m}{2}[2a_1+(m - 1)d]$. Here, $m=n - 1$, $a_1 = 5$, and $d = 4$. The sum of the first $(n - 1)$ terms of the sequence of differences is $S_{n - 1}=\frac{(n - 1)}{2}[2\times5+(n - 1 - 1)\times4]=\frac{(n - 1)}{2}(10+4(n - 2))=\frac{(n - 1)}{2}(10 + 4n-8)=\frac{(n - 1)}{2}(4n + 2)=(n - 1)(2n + 1)=2n^2 - n - 1$. Since $a_1 = 1$, the $n$th term $a_n=1+(2n^2 - n - 1)=2n^2 - n$.

Answer:

$a_n=2n^2 - n$