Given that the first $5$ terms of a sequence, $T_n$, are $-2, 2, 6, 10, 14, \dots$, where&nbsp

Question

Given that the first $5$ terms of a sequence, $T_n$, are $-2, 2, 6, 10, 14, \dots$, where $n = 1, 2, 3, 4$ and $5$. Find a formula for the $n^{th}$ term of the sequence, $T_n$, in terms of $n$. Hence find $T_0$. (Hint: substitute $n = 0$ into the formula of the general term of $T_n$.) In the grid below, plot $T_n$ against $n$, i.e., draw the points of which the coordinates are in the form of $(n, T_n)$, where $n = 1, 2, 3, 4$ and $5$. Hence, draw a straight line that passes through these points. Using your graph, find the gradient, $m$, and the $T_n$-intercept, $c$, of the straight line. Hence write down the equation of the straight line in the form of $T_n = mn + c$. Based on your answers above, determine whether whether the point $\left( \frac{1}{2},-4 \right)$ lies on the graph of the straight line or not, and $-4$ is a term of the sequence $T_n$ or not. Explain your answers.

Accepted Answer

(i) $T_n=4n-6$, $T_0=-6$ (ii) Straight line through $(1,-2)$, $(2,2)$, $(3,6)$, $(4,10)$, $(5,14)$ (iii) $m=4$, $c=-6$, $T_n=4n-6$ (iv)(a) Yes (iv)(b) No

Number Patterns

Number Patterns

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