Question

Show that $ {t_n} $ for the A.P.: $ 13,\,\,18,\,\,23,\,\,28........... $ is $ 5n + 8 $

Answer 1

Hint: First from the given A.P. series we find value of ‘a’ first term and ‘d’ common difference and then use these values of ‘a’ and ‘d’ in nth term formula to find its nth term of the given A.P. series.
 $ {t_n} = a + \left( {n - 1} \right)d, $ where ‘a’ is the first term and ‘d’ is the common difference of the given A.P. and ‘n’ stands for nth term of the given series.

Complete step-by-step answer:
Given A.P. is $ 13,\,\,18,\,\,23,\,\,28........... $
From above series we have
First term of the series ‘a’ = $ 13 $
Also, we know that in A.P. every two consecutive terms differ by a number named as a common difference denoted by symbol (d).
Hence, to find the value of common difference (d) we should calculate the difference of any two consecutive terms of the given A.P. series.
Therefore, d = $ 18 - 13\,\,or\,\,\,23 - 18 = 5 $
Now, to find nth term of the given A.P. series we used nth term formula which is given as
 $ {t_n} = a + \left( {n - 1} \right)d $
Substituting values in above mentioned formula we have
 $ {t_n} = 13 + \left( {n - 1} \right)\left( 5 \right) $
Simplifying right hand side of the above equation we have
 $
  {t_n} = 13 + 5n - 5 \\
   \Rightarrow {t_n} = 5n - 8 \\
  $
Therefore, the nth term of the given A.P. is $ 5n - 8 $ .
Hence, from above we see that nth term which we have derived after simplification matches with the given nth term.

Note: Nth term of general term of any A.P. given series can be obtained by using formula and substituting given values of ‘a’ and ‘d’ but not ‘n’ as for nth there is no fixed value of ‘n’ is known. Hence we keep ‘n’ as it is in formula to get an answer in terms of ‘n’.