Finding the position number of a term and number of terms in Arithmetic Progression
In this article, we look at how you can determine the position number of a term in an Arithmetic Progression (A.P.) or Linear Sequence.
Douglas Tawiah Dwumor

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- Given common difference, first term the position number of any term can be obtained if the term is known
- To find the position number of a term, you make n the subject.
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If a term in a linear sequence is given, we can find the position number (n) of the term. That’s, whether the term is 4th or 28th in the sequence. Note that n represents the position number of any term in the A.P. or linear sequence.
Example 1:
In the A.P., $ 5, 11, 17,… $, what term is 101?
Solution
$a=5, d=11-5=6, n=?$
$U_n=a + (n-1)d$
$101 = 5 + (n-1)(6)$
$101=5 + 6n-6$
$101 = 6n -1$
$102 = 6n$
$n=17$
Thus, 101, is the $17th$ term.
Example 2:
Given the sequence $-9, -6 \frac{1}{2}, -3 \frac{1}{2},…$, where does the term $18 \frac{1}{2}$ belong?
Solution
$a=-9, d=-7.5-(-9)=2.5=6, n=? $
$Un=a + (n-1)d$
$18.5 = -9 + (n-1)(2.5)$
$18.5 =-9 + 2.5n -2.5$
$18.5 = -11.5 + 2.5n$
$30 = 2.5n$
$n=12$
Hence, $18 \frac{1}{2}$ is the $12th$ term in the sequence.
Finding number of terms in a sequence
If we can find the position number (n) of any term in an A.P. , then we can find the position number of the last term too in a finite sequence.
The position number of the last term also gives the total number of terms in a finite sequence.
That’s, given the general term
$Un= a + (n-1)d,$
if $Un=l$ where $l$ is last term,
then $n=$number of terms in the sequence
That’s,
$l=a+ (n-1)d$
where $n=$position number of last term or total number of terms in the finite sequence.
Example 1:
Find the number of terms in the linear sequence $2,-9,-20,…-141$.
Solution
$a=2, d=-9-2=-11, l=-141, n=?$
$l=a + (n-1)d$
$-141=2 +(n-1)(-11)$
$-141= 2 -11n + 11$
$-141 -13=-11n$
$-154=-11n$
$ n=14$
The position number of the last term, $-141$, is $14$. Hence, there are $14$ terms in the linear sequence.
Example 2:
How many terms are in the linear sequence $5,8,11,…….122$ ?
Solution
$a=5, d=8-5=3, l=122, n=?$
$l=a + (n-1)d$
$122=2 +(n-1)(3)$
$122= 2 +3n -3$
$122= -1+3n$
$123=3n$
$n=41$
The position number of the last term, 122, is $40$. Hence, there are $40$ terms in the linear sequence.
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