Question

# If sum of n terms of an A.P is ${\text{3 - 4n}}$, find, the common difference.

Hint: As we know that nth term of an A.P can be calculated as ${{\text{T}}_{\text{n}}}{\text{ = }}{{\text{S}}_{\text{n}}}{\text{ - }}{{\text{S}}_{{\text{n - 1}}}}$. And so as we know ${{\text{S}}_{\text{n}}}$ as per given in the question. So, we calculate ${{\text{S}}_1}{\text{,}}{{\text{S}}_2}$and their difference will give us the second term, and then on subtraction of first term from second term we get our answer.

As per the given , ${{\text{S}}_{\text{n}}}{\text{ = 3 - 4n}}$
So, we first calculate ${{\text{S}}_1}{\text{ and }}{{\text{S}}_2}$,
${{\text{S}}_1}{\text{ = 3 - 4(1) = - 1,}} \\ {{\text{S}}_2} = 3 - 4(2) = - 5 \\$
Now we find ${{\text{S}}_2} - {{\text{S}}_1}$,
$\Rightarrow {{\text{S}}_2} - {{\text{S}}_1} = - 5 - ( - 1) = - 4$
So we have ${{\text{S}}_1}{\text{ = - 1 = }}{{\text{a}}_1}$ and ${{\text{S}}_2}{\text{ = - 4 = }}{{\text{a}}_2}$.
So, difference will be ${{\text{a}}_{\text{2}}}{\text{ - }}{{\text{a}}_{\text{1}}}{\text{ = - 4 - ( - 1) = - 3}}$
Hence , ${\text{ - 3}}$ is our required answer.

Note: In mathematics, an arithmetic sequence is a sequence of numbers such that the difference between the consecutive terms is constant.
Properties of Arithmetic Progressions
1)If the same number is added or subtracted from each term of an A.P, then the resulting terms in the sequence are also in A.P with the same common difference.
2)If each term in an A.P is divided or multiplied with the same non-zero number, then the resulting sequence is also in an A.P.