question 2 predict the nth term in the following patterns. a) 3, 7, 11, 15, ... b) 0, 2, 6, 12, ... c) 1, 6…

Explanation:

Step1: Analyze sequence a

The sequence 3, 7, 11, 15,... is an arithmetic - sequence with a first term $a_1 = 3$ and a common difference $d=4$. The formula for the $n$th term of an arithmetic sequence is $a_n=a_1+(n - 1)d$. Substituting $a_1 = 3$ and $d = 4$ into the formula, we get $a_n=3+(n - 1)\times4=3 + 4n-4=4n - 1$.

Step2: Analyze sequence b

The sequence 0, 2, 6, 12,... The differences between consecutive terms are 2, 4, 6,... which form an arithmetic sequence. The $n$th term of this sequence can be found by observing that $a_n=n(n - 1)$. When $n = 1$, $a_1=1\times(1 - 1)=0$; when $n = 2$, $a_2=2\times(2 - 1)=2$; when $n = 3$, $a_3=3\times(3 - 1)=6$; when $n = 4$, $a_4=4\times(4 - 1)=12$.

Step3: Analyze sequence c

The sequence 1, 6, 15, 28,... The differences between consecutive terms are 5, 9, 13,... which is an arithmetic sequence with a common difference of 4. The $n$th term formula is $a_n=2n^2 - n$. When $n = 1$, $a_1=2\times1^2-1=1$; when $n = 2$, $a_2=2\times2^2-2=6$; when $n = 3$, $a_3=2\times3^2-3=15$; when $n = 4$, $a_4=2\times4^2-4=28$.

Answer:

a) $a_n = 4n-1$ b) $a_n=n(n - 1)$ c) $a_n=2n^2 - n$