Question

The $ {6^{th}} $ and $ {17^{th}} $ terms of an A.P. are $ 19 $ and $ 41 $ respectively, find the $ {40^{th}} $ term.

Answer 1

Hint: Here, we are given the arithmetic progression for two terms, so will place the value in the standard equation of the arithmetic progression and will find the unknowns first term (a) and the difference (d) between the terms and accordingly find the $ {40^{th}} $ term. Use the equation –
 $ {a_n} = a + (n - 1)d $ Where, a is the first term, d is the difference between the two terms, n is the number of terms and $ {a_n} $ is the nth term.

Complete step-by-step answer:
The $ {6^{th}} $ and $ {17^{th}} $ terms of an A.P. are $ 19 $ and $ 41 $ respectively.
Convert the above statement in the form of mathematical expressions.
 $
  19 = a + (6 - 1)d \\
  19 = a + 5d\;{\text{ (1)}} \;
  $
 $
  41 = a + (17 - 1)d \\
  41 = a + 16d\;{\text{ (2)}} \;
  $
Subtract equation $ (1) $ from $ (2) $
 $
   \Rightarrow 41 - 19 = a - a + 16d - 5d \\
   \Rightarrow 22 = 11d \;
  $
When the number in the multiplication with the term changes its side, then it goes in the division.
 $
   \Rightarrow d = \dfrac{{22}}{{11}} \\
   \Rightarrow d = 2 \;
  $
Place the value of “d” in the equation $ (1) $
 $
  19 = a + 5(2) \\
   \Rightarrow 19 = a + 10 \;
  $
When the term changes its sides, sign of the term is also changed. Positive number changes to negative and vice-versa.
 $
   \Rightarrow a = 19 - 10 \\
   \Rightarrow a = 9 \;
  $
Now, the required $ {40^{th}} $ term –
 $ {a_n} = a + (n - 1)d $
Place the values of “a” , “n” and “d”
 $ {a_{40}} = 9 + (40 - 1)(2) $
Simplify the above equation –
 $
  {a_{40}} = 9 + (39)(2) \\
  {a_{40}} = 9 + 78 \\
  {a_{40}} = 87 \;
  $
Therefore, the required answer - The $ 40th $ term is $ 87. $
So, the correct answer is “87”.

Note: Remember the difference between two types of sequences and series.
Arithmetic progression (A.P.) and
 Geometric Progression (G.P.)
In arithmetic progression, the difference between the numbers is constant in the series whereas the geometric progression is the sequence in which the succeeding element is obtained by multiplying the preceding number by the constant and the same continues for the series. The ratio between the two remains the same.