Question

How do you find the nth term of the sequence $ 1,3,6,10,15,.... $

Answer 1

Hint: In this question, an arrangement of five numbers is given to us and we have to find the nth term of this arrangement. We must find some pattern that is followed by an arrangement to find the next terms of the arrangement. We see that the numbers are getting bigger as we move towards the right side so the difference between any two numbers and it is also increasing on moving towards the right. Applying some arithmetic operations on any two sets of consecutive terms, we can find the pattern and thus the nth term.

Complete step-by-step answer:
The difference between the consecutive pairs of the term is –
 $ 2,3,4,5,... $
The difference between the consecutive pair of the arrangement obtained is –
 $ 1,1,1,... $
Now an arrangement of constant terms is reached so, we can find the formula for the nth term as follows –
 $
  {a_n} = \dfrac{1}{{0!}} + \dfrac{2}{{1!}}(n - 1) + \dfrac{3}{{2!}}(n - 1)(n - 2) \\
   \Rightarrow {a_n} = 1 + 2(n - 1) + \dfrac{1}{2}(n - 1)(n - 2) \\
   \Rightarrow {a_n} = 1 + 2n - 2 + \dfrac{1}{2}{n^2} - \dfrac{3}{2}n + 1 \\
   \Rightarrow {a_n} = \dfrac{1}{2}{n^2} + \dfrac{1}{2}n \\
   \Rightarrow {a_n} = \dfrac{{n(n + 1)}}{2} \;
  $
Hence the nth term of $ 1,3,6,10,15,.... $ is given as $ \dfrac{{n(n + 1)}}{2} $
So, the correct answer is “ $ \dfrac{{n(n + 1)}}{2} $ ”.

Note: A sequence or series is defined as an arrangement of numbers that follows some pattern as in this question. When we are not able to find a simple pattern in the arrangement of numbers, we use the above-mentioned method, that is, we keep finding the series of differences between the consecutive terms until we reach a constant arrangement and then we apply the formula for finding the nth term.