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How do you write an equation for the nth term of the arithmetic sequence: $ - 3, - 5, - 7, - 9,....?$

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Last updated date: 18th Jun 2024
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Answer
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Hint: An Arithmetic Progression (AP) is the sequence of numbers in which the difference of two successive numbers is always constant.

The standard formula for Arithmetic Progression is: ${t_n} = a + (n - 1)d$
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 we will find the first term and the difference between the two successive terms and will place it in the standard equation.

Complete step-by-step solution:
Referring the given sequence, we get
First term, $a = - 3$
Difference, $d = - 5 - ( - 3)$
When the product of two minus sign gives, resultant value in plus.
 $ \Rightarrow d = - 5 + 3$
While simplifying one negative term and one positive term, you have to do subtraction and give sign of bigger numbers.
$ \Rightarrow d = - 2$
Now, nth term of the given arithmetic expression would be-
$\Rightarrow{t_n} = a + (n - 1)d$
Place values of the first term and the common difference in the above equation.
$\Rightarrow{t_n} = ( - 3) + (n - 1)( - 2)$
Always remember the product of one positive term and one negative term gives resultant value in negative.
$\Rightarrow {t_n} = ( - 3) - 2(n - 1)$
When there is a negative sign outside the bracket then all the terms inside the bracket changes when opened.
$\Rightarrow {t_n} = ( - 3) - 2n + 2$
Make a pair of like terms in the above equation.
$\Rightarrow {t_n} = \underline {( - 3) + 2} - 2n$
Simplify the above equation.
$\Rightarrow {t_n} = ( - 1) - 2n$
This is the required solution.

Note: Know the difference between the arithmetic and geometric progression and apply the concepts accordingly. In arithmetic progression, the difference between the numbers is constant in the series whereas, the geometric progression is the sequence in which the succeeding element is obtained by multiplying the preceding number by the constant and the same continues for the series. The ratio between the two remains the same.