Using Tₙ — a Term & Which Term

Section 1 of 2

Finding a Particular Term

Rule first, then substitute

To find a far-off term (like $T_{30}$), first build $T_n$, then put in that value of $n$.

Worked example — find and hence use Tn

For $3, 11, 19, 27, \ldots$ find $T_n$ and hence $T_{30}$.

$a = 3$, $d = 8$; $T_n = 3 + 8(n-1) = 8n - 5$

$T_{30} = 8(30) - 5 = 235$

$T_n = 8n - 5,\ T_{30} = 235$

You try

For $-3, 6, 15, \ldots$ find $T_n$ and hence $T_{50}$.

$a = -3$, $d = 9$.

$T_n = -3 + 9(n-1) = 9n - 12$

$T_{50} = 9(50) - 12 = 438$

$438$

You try

For $15, 13, 11, \ldots$ find $T_n$ and hence $T_{20}$.

$a = 15$, $d = -2$.

$T_n = 15 - 2(n-1) = 17 - 2n$

$T_{20} = 17 - 2(20) = -23$

$-23$

Section 2 of 2

Which Term Has That Value?

Solve for n

To find which term equals a value, set $T_n$ equal to it and solve for $n$.

Worked example — which term

Which term of $1, 3, 5, 7, \ldots$ has the value $121$?

$a = 1$, $d = 2$; $T_n = 2n - 1$

$2n - 1 = 121 \Rightarrow 2n = 122$

$n = 61$ (the $61$st term)

$n = 61$

Worked example — a term and a value

$T_n = 6n - 5$. Find (i) $T_7$; (ii) which term has value $67$.

(i) $T_7 = 6(7) - 5 = 30$

(ii) $6n - 5 = 67 \Rightarrow 6n = 72 \Rightarrow n = 12$

$T_7 = 30,\ n = 12$

You try

Which term of $5, 8, 11, \ldots$ has value $272$?

$T_n = 3n + 2$; set $= 272$.

$3n + 2 = 272 \Rightarrow 3n = 270 \Rightarrow n = 90$

$n = 90$

You try

Which term of $3, 9, 15, \ldots$ has value $117$?

$T_n = 6n - 3$; set $= 117$.

$6n - 3 = 117 \Rightarrow 6n = 120 \Rightarrow n = 20$

$n = 20$

You try

$T_n = 3n + 9$. What term has a value of $318$?

$3n + 9 = 318$.

$3n = 309 \Rightarrow n = 103$

$n = 103$

You try

For $-3, -1, 1, \ldots$ find $T_n$, then which term has value $1075$.

$a = -3$, $d = 2$, $T_n = 2n - 5$.

$T_n = 2n - 5$

$2n - 5 = 1075 \Rightarrow 2n = 1080 \Rightarrow n = 540$

$n = 540$

That’s Class 3.

Finding a term $T_k$ and finding which term equals a value. Class 4: consecutive terms.