7.2General Term of a Sequence

The general term $T_n$ gives any term straight from its position, so the 500th needs no list.

A sequence is a function of position: each $n$ gives exactly one $T_n$ .

Evaluating the

k

th term

\displaystyle \text{To find } T_k \text{, substitute } n = k \text{ into the formula for } T_n

Evaluating terms from a formula

Given $\displaystyle T_n = 4n - 1$ , find (i) $\displaystyle T_4$ and (ii) $\displaystyle T_6 - T_2$ .

Show solution▾

(i)

T_4 = 4(4) - 1

\displaystyle = 16 - 1

\displaystyle = 15

(ii)

T_6 = 4(6) - 1

\displaystyle = 24 - 1

\displaystyle = 23

\displaystyle T_2

\displaystyle = 4(2) - 1

\displaystyle = 8 - 1

\displaystyle = 7

\displaystyle T_6 - T_2

\displaystyle = 23 - 7

\displaystyle = 16

Now you try

Given $T_n = 6n + 4$ , find $T_1$ .

Show solution▾

\displaystyle T_n

\displaystyle = 6n + 4

\displaystyle T_1

\displaystyle = 6(1) + 4 \quad \scriptsize\textit{substitute } n = 1

\displaystyle = 6 + 4

\displaystyle = 10

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Finding which term equals a given value

Given $\displaystyle T_n = 3n + 8$ , find the value of $\displaystyle n$ for which $\displaystyle T_n = 35$ .

Show solution▾

\displaystyle 3n + 8

\displaystyle = 35

\displaystyle 3n

\displaystyle = 27

\displaystyle n

\displaystyle = 9

The 9th term equals 35.

Check:

T_9 = 3(9) + 8 = 35

✓

Reverse problem: find $n$ given $T_n$

The $\displaystyle n$ th term of a sequence is $\displaystyle T_n = 5n - 3$ . Find the value of $\displaystyle n$ for which $\displaystyle T_n = 72$ .

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Set

T_n = 72

\displaystyle 5n - 3

\displaystyle = 72

\displaystyle 5n

\displaystyle = 75

\displaystyle n

\displaystyle = 15

Check:

T_{15} = 5(15) - 3 = 75 - 3 = 72

✓

Now you try

The nth term of a sequence is $T_n = n^2 - 5n - 2$ . Find the smallest value of n for which $T_n > 12$ .

Show solution▾

T_n = n^2 - 5n - 2

n^2 - 5n - 2 > 12 \quad \scriptsize\textit{set up inequality}

n^2 - 5n - 14 > 0 \quad \scriptsize\textit{rearrange}

\displaystyle T_{7}

\displaystyle = 12 \quad \scriptsize\textit{check boundary}

\displaystyle T_{8}

\displaystyle = 22 > 12 \quad \scriptsize\textit{first term exceeding threshold}

\displaystyle n

\displaystyle = 8

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Reverse problem: find $a$ and $b$ from two known terms

A sequence has general term $\displaystyle T_n = an + b$ . Given $\displaystyle T_3 = 14$ and $\displaystyle T_7 = 30$ , find $\displaystyle a$ and $\displaystyle b$ , and hence state the general term.

Show solution▾

Substituting

n = 3

3a + b = 14

\quad

(1)

Substituting

n = 7

7a + b = 30

\quad

(2)

Subtract (1) from (2):

4a = 16

a = 4

Substitute into (1):

3(4) + b = 14

\displaystyle 12 + b

\displaystyle = 14

\displaystyle b

\displaystyle = 2

\displaystyle T_n

\displaystyle = 4n + 2

Check:

T_3 = 4(3) + 2 = 14

✓,

T_7 = 4(7) + 2 = 30

✓

Recognising common general terms

Write the general term for each sequence: (a) $\displaystyle 5, 10, 15, 20, 25, \ldots$ (b) $\displaystyle 1, 4, 9, 16, 25, \ldots$ (c) $\displaystyle 2, 8, 18, 32, 50, \ldots$

Show solution▾

(a) Positive multiples of 5:

T_n = 5n

Check:

T_1 = 5(1) = 5

✓,

T_4 = 5(4) = 20

✓

(b) Perfect squares:

T_n = n^2

Check:

T_3 = 3^2 = 9

✓,

T_5 = 5^2 = 25

✓

T_n = 2n^2

Check:

T_2 = 2(2^2) = 8

✓,

T_5 = 2(5^2) = 50

✓

7.2General Term of a Sequence

Evaluating terms from a formula

Finding which term equals a given value

Reverse problem: find nnn given TnT_nTn​

Reverse problem: find aaa and bbb from two known terms

Recognising common general terms

Practise this

Reverse problem: find $n$ given $T_n$

Reverse problem: find $a$ and $b$ from two known terms