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 $T_n = 5n - 4$ , find $T_{7}$ .

Show solution▾

\displaystyle T_n

\displaystyle = 5n - 4

\displaystyle T_{7}

\displaystyle = 5(7) - 4 \quad \scriptsize\textit{substitute } n = 7

\displaystyle = 35 - 4

\displaystyle = 31

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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 $\displaystyle T_n = 35$ .

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\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

Given $T_n = 2n^2 - 5n + 3$ , find the positive integer value of n such that $T_n = 15$ .

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2n^2 - 5n + 3 = 15

We want

T_n = 15

, so try values of n in turn.

\displaystyle T_{2}

\displaystyle = 1

\displaystyle T_{3}

\displaystyle = 6

T_{4} = 15

, which is the value we want.

\therefore n = 4

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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▾

Going from

T_3

T_7

7 - 3 = 4

steps along the sequence.

Over those steps the value rises by

30 - 14 = 16

T_n = an + b

, each step adds

a

, so

a = \dfrac{16}{4}

a = 4

Substituting

n = 3

4(3) + 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