Question

Find the $ 30th $ term of the sequence: $ \dfrac{1}{2},1,\dfrac{3}{2}........ $

Answer 1

Hint: First of all we will find the type of given sequence. Sequences follow some basic pattern and accordingly we will use the standard formula to get the specific term for the sequence and will simplify for the resultant required value.

Complete step-by-step answer:
Given Sequence: $ \dfrac{1}{2},1,\dfrac{3}{2}........ $
First term, $ a = \dfrac{1}{2} $
And common difference, $ d = 1 - \dfrac{1}{2} = \dfrac{1}{2} $
Now, the given sequence here is Arithmetic progression where the common difference between all the terms remains the same.
The standard formula for the arithmetic progression is,
 $ {T_n} = a + (n - 1)d $
Place the identified values in the above equation. Place $ n = 30 $ $ $
 $ {T_{30}} = \dfrac{1}{2} + (30 - 1)\dfrac{1}{2} $
Simplify the above expression –
 $ {T_{30}} = \dfrac{1}{2} + (29)\dfrac{1}{2} $
Simplify the above equation –
 $ {T_{30}} = \dfrac{1}{2} + \dfrac{{29}}{2} $
When denominators are the same, numerators are combined.
 $ {T_{30}} = \dfrac{{29 + 1}}{2} $
Simplify the above expression finding the sum of terms in the numerator
 $ {T_{30}} = \dfrac{{30}}{2} $
Find the factors of the term in the numerator.
 $ {T_{30}} = \dfrac{{15 \times 2}}{2} $
Common factors from the numerator and the denominator cancel each other and therefore remove from the numerator and the denominator.
 $ {T_{30}} = 15 $
Hence, the $ 30th $ for the given series is $ 15 $
So, the correct answer is “ $ 15 $ ”.

Note: Do not get confused, in identifying the given sequence. Basically, there are two types of sequences. Arithmetic Progression and the geometric progression. In arithmetic sequences the common difference between the two terms always remains the same while in geometric progression the ratio between the two consecutive terms remains the same. Be good in basic four mathematical operations to solve.