Question

How do you write an explicit rule for the sequence $ 3,5,7,9... $ ?

Answer 1

Hint: Each sequence follows a pattern and based on this pattern, the sequences are named differently. To find the $ {n^{th}} $ term of any sequence, we use a formula called the explicit formula. In simple words, explicit is called exact or definite, as the formula for finding the $ {n^{th}} $ term gives the exact value of the term at $ {n^{th}} $ place, it is called an explicit formula. The given sequence is an Arithmetic Sequence, so by using the formula for finding the $ {n^{th}} $ term of an A.P. we can find out the explicit rule for the given sequence.

Complete step-by-step answer:
The given sequence is $ 3,5,7,9... $
The first term of the given sequence is 3 so $ a = 3 $
The common difference of the given sequence is calculated as –
 $
  d = 5 - 3 = 7 - 5 \\
   \Rightarrow d = 2 \;
  $
The $ {n^{th}} $ term of the arithmetic sequence is given by the formula $ {a_n} = a + (n - 1)d $
We know the values of a and d, so we get –
 $ {a_n} = 3 + (n - 1)2 $
Hence, the explicit rule for the sequence $ 3,5,7,9... $ is $ {a_n} = 3 + (n - 1)2 $ .
So, the correct answer is “ $ {a_n} = 3 + (n - 1)2 $ ”.

Note: The main two types of sequences are Arithmetic Sequence and Geometric Sequence. An Arithmetic progression is a progression or sequence of numbers such that the difference between any two consecutive numbers is constant. The first term of the arithmetic sequence is the term from which the sequence begins and the difference between any two consecutive terms is called the common difference of that arithmetic progression. Using the definition of these two terms, we found the first term and the common difference of the given sequence. The formula for finding the $ {n^{th}} $ term of an A.P. can be obtained if we keep on adding the common difference for $ (n - 1) $ times.