Question

Considering the sequence as Arithmetic Progression (AP), explain how to find the $ {20^{th}} $ term of any sequential series.

Answer 1

Hint: Arithmetic Progression may contain any sequence having numeral, algebraic variables, etc… For determining any of the terms in a complex series of AP we can find it with the help of the formula $ {t_n} = a + (n - 1)d $ which contains the first term, common difference and required number ‘n’ particularly.

Complete step-by-step answer:
As per as the mathematical part is concerned, arithmetic progression [AP] (also called as an arithmetic sequence) is the series of the numbers, variables, etc… in which there is a difference between all the consecutive terms in the respective series is constant!
As a result, from the definition the equation seems that
 $ [{t_n} = a + (n - 1)d] $ … (i)
Where, ‘a’ is the first term of the sequence or series,
‘n’ is the required term in the respective series or sequence,
‘d’ is the common difference of the arithmetic series,
Mathematically, expressed as $ [d = {t_2} - {t_1}] $ where, $ {t_2} $ is second term and \[{t_1}\] is first term of the series respectively
Let’s take an example of the sequence existing AP for more clarifications,
Q. $ 2,4,6,8,....100 $ Consists of AP. Find the $ {20^{th}} $ term.
As a result, simplifying the equation by considering the parameters explained above, we get
 $ 2,4,6,8,....100 $ is the given sequence where,
 $
  a = 2 \\
  d = {t_2} - {t_1} = 4 - 2 = 2 \;
  $ and,
Since, the required term is $ {20^{th}} $ term.
 $ \therefore n = 20 $
Now, substituting all the values in the equation (i), we get
\[{t_{20}} = 2 + (20 - 1) \times 2\]
Solving the equation mathematically, we get
\[
  {t_{20}} = 2 + 19 \times 2 = 2 + 38 \\
  {t_{20}} = 40 \;
 \]
Hence, the $ {20^{th}} $ term of the AP series $ 40 $ respectively.

Note: Remember the formulae for $ {t_n} = a + (n - 1)d $ which gives the exact term in the sequence of Arithmetic Progression (AP). We can also take the sum of all the terms of the AP sequence by using the formula $ {S_n} = \dfrac{n}{2}\left[ {2a + (n - 1)d} \right] $ respectively. One must keenly observe the desired sequence of the AP to determine the respective solution!