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# In an A.P. if $a = 13,\;{{\text{T}}_{15}} = 55,$ find $d.$

Last updated date: 28th Feb 2024
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Hint:Use the formula for finding the nth term of an arithmetic progression when its first term (a) and common difference (d) are given, and then put the given data in that equation or formula, you will then get one equation and one variable. Just solve the equation for that variable you will get the required common difference.
nth term of an A.P. with “a” as its first term and “d” be its common difference is given as follows:
${{\text{T}}_n} = a + (n - 1)d$

Complete step by step solution:
To find “d”, let us first understand the meaning of all terms in the question.
The question is saying in an Arithmetic Progression (A.P.) if its first term and the fifteenth term is given $(a = 13,\;{\text{T}} = 55)$ then we have to find its common difference $(d)$, we will proceed in this problem with the arithmetic formula for the nth term, which is given as follows
${{\text{T}}_n} = a + (n - 1)d,\;where\;{{\text{T}}_n},\;a\;{\text{and}}\;d$ are nth term, first term and the common difference of the arithmetic series.
According to the question,
$\Rightarrow 55 = 13 + (15 - 1)d \\ \Rightarrow 55 = 13 + 14d \\ \Rightarrow 14d = 55 - 13 \\ \Rightarrow 14d = 42 \\ \Rightarrow d = \dfrac{{42}}{{14}} \\ \Rightarrow d = 3 \\$
Therefore the required common difference of the Arithmetic Progression, whose first term is $13$ and fifteenth term is $55$ is equals to $3$
Formula used: Arithmetic progression formula for nth term: ${{\text{T}}_n} = a + (n - 1)d$
Additional information: Nature of an A.P. depends upon its common difference that is A.P. is increasing or decreasing according to the common difference is positive or negative

Note: Common difference is the fixed number or constant number that is further added and added to a number to get an arithmetic progression of numbers. Common difference of an arithmetic progression can be achieved by subtracting any term of the arithmetic progression from its consecutive next term.