Question

What is the nth term of the arithmetic sequence $ 1,3,5,7,9,11,....? $
A. $ n + 1 $
B. $ n - 1 $
C. $ 2n + 1 $
D. $ 2n - 1 $

Answer 1

Hint: An Arithmetic Progression (AP) can be expressed as the sequence of numbers in which the difference of two successive numbers is always constant.
The standard formula for Arithmetic Progression is: $ {a_n} = a + (n - 1)d $
Where $ {t_n} = $ nth term in the AP
 $ a = $ First term of AP
 $ d = $ Common difference in the series
 $ n = $ Number of terms in the AP
Here, by identifying the first term, common difference and nth term will find the required term.

Complete step-by-step answer:
First term, $ a = 1 $
Common difference, $ d = 3 - 1 = 2 $
Now, the nth term for the given sequence is given by
 $ {a_n} = a + (n - 1)d $
Place the identified values in the above equation –
 $ {a_n} = 1 + (n - 1)2 $
Simplifying the above expression multiplying term outside the bracket with the terms inside the bracket.
 $ {a_n} = 1 + 2n - 2 $
Combine the like terms in the above equation –
 $ {a_n} = 2n\underline { - 2 + 1} $
When you add smaller positive number to the larger negative number, you have to do subtraction and give sign of the negative to the resultant value
 $ {a_n} = 2n - 1 $
From the given multiple choices, the option D is the correct answer.
So, the correct answer is “Option D”.

Note: Know the difference between the arithmetic sequence and the geometric sequence and know its standard formula. In arithmetic sequence, the difference between the two terms remains the same and in geometric sequence the ratio between the two remains the same. Be good in simplifying the equation while keeping the sign convention in mind while combining the terms with different signs.