Question

# Find the ${10^{th}}$ term from the last term of the AP: $8,10,12.....,126$

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Hint: Arithmetic Progression (AP) is a sequence of numbers in order in which the difference of any two consecutive numbers is a constant value. For example, the series of natural numbers: $1,{\text{ }}2,{\text{ }}3,{\text{ }}4,{\text{ }}5,{\text{ }}6,$is an AP, which has a common difference between two successive terms (say 1 and 2) equal to 1 (2 -1). Even in the case of odd numbers and even numbers, we can see the common difference between two successive terms will be equal to 2.
Definition 1: A mathematical sequence in which the difference between two consecutive terms is always a constant and it is abbreviated as AP.
Definition 2: An arithmetic sequence or progression is defined as a sequence of numbers in which for every pair of consecutive terms, the second number is obtained by adding a fixed number to the first one.
In arithmetic progression (A.P) series the first term is denoted by ‘$a$’ and the common difference is denoted by ‘$d$’ and ‘$n$’ is a number of terms. ‘${a_n}$’ is last term. Here in the series given the value of ‘$a$’ is $8$and ‘$d$’ is$2$.

$8,10,12,....126$
Here first term $a$=$8$
Common difference $=$$10 - 8 = 2$
Last term ($l$)$= 126$
Number of terms$= 10$
Now ${n^{th}}$ term from end using formula
$l - (n - 1)d$
$= 126 - (10 - 1)2$
$= 126 - 9 \times 2$
$= 108$
Note: We can also find ${n^{th}}$ term from beginning by using formula ${a_n} = a + (n - 1)d$ and also sum of $n$ terms by using formula ${S_n} = \dfrac{n}{2}(2a + (n - 1)d)$ or ${S_n} = \dfrac{n}{2}(a + l)$ if last term is given.