Question 5 of 51beginner💡 UnderstandEssay2 marks

Why is it useful to have an explicit formula for the nthn^{th} term of a sequence?

Correct Answer

Having an explicit formula for the nthn^{th} term of a sequence is highly useful because it allows us to find any term directly. By just substituting the appropriate value of nn, we can quickly calculate the 20th20^{th}, 53rd53^{rd}, or 300th300^{th} term without having to know the values of the previous terms.

Additionally, an explicit formula helps us check if a certain number is a term of the sequence. We can also use it to find the exact position of a given term by solving the equation for nn. For example, using un=2n−1u_n = 2n - 1, we can find that 137 is the 69th69^{th} term by solving 2n−1=1372n - 1 = 137.

Exercise: Think and Reflect | Q: 1 | (Page 4)
For More Understanding

Explanation

The textbook highlights two main advantages of an explicit formula over a recursive one. First, it provides direct access to any term in the sequence just by substituting the position number, eliminating the need to calculate all preceding terms. Second, it allows working backwards to verify if a number belongs to the sequence and to determine its exact position by solving a simple equation.

Solution Steps

  1. Step 1: State that an explicit formula allows direct calculation of any term by substituting 'n' without needing previous terms.

  2. Step 2: State that it helps check if a number is a term of the sequence and find its position by solving for 'n'.

PreviousNext