Question

How do you write the expression for the $n^{th}$ term of the sequence given
$1,\dfrac{1}{4},\dfrac{1}{9},\dfrac{1}{16},\dfrac{1}{25},...$?

Answer 1

To solve these types of questions, first we need to find the common pattern that each of the terms follows. By identifying this pattern, we can write each term in a general form. By using this general term formula, we can write the expression for the $n^{th}$ term of the series.

Complete step by step answer:
The given sequence is $1,\dfrac{1}{4},\dfrac{1}{9},\dfrac{1}{16},\dfrac{1}{25},...$. We need to find the expression for the $n^{th}$ term of this sequence, to do this we will first find the general pattern that each term follows. The first term of the sequence is 1, we can also write this term as $\dfrac{1}{1}$. The denominator of the term is 1, we know that 1 is square of itself. The second term is $\dfrac{1}{4}$, the denominator of the term is 4. We know that 4 is a square of 2. Similarly, the denominator of terms at the position 3, 4, and 5 are 9, 16, and 25. These are the squares of the position number of the term.
Thus, the pattern that each term follows is that the denominator of each term is the square of the number of positions.
Using this, we can express the $n^{th}$ term of the sequence as $\dfrac{1}{{{n}^{2}}}$.

Note: While solving the questions based on sequence, it is always suggested to find the expression of the $n^{th}$ term of the series. We can derive different useful results using this expression as the sum of n terms of the series.