Question

the sum of the first n terms of an ap is given by sn=3n2-n.find it's nth term
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Answer 1

aₙ = 6n - 2

Given

The sum of the first n terms of an AP is given by Sₙ = 3n² - n

To Find

nth term

Knowledge required

$\bf \pink{\bigstar\ \; a_n=a+(n-1)d}\\\\\bf \green{\bigstar\ \; S_n=\dfrac{n}{2}(2a+(n-1)d)}$

Solution

$\rm S_n=3n^2-n\\\\\rm \to S_1=3(1)^2-(1)\\\\\to \rm a=3-1=2\\\\\bf \to a=2$

Now ,

$\rm S_2=3(2)^2-(2)\\\\\to \rm S_2=3(4)-2\\\\\to \rm S_2=10\\\\\to \rm a_1+a_2=10\\\\\to \rm a+a+d=10\\\\\to \rm 2a+d=10\\\\\to \rm 2(2)+d=10\\\\\to \bf d=6$

So , nth term of AP ,

$\implies \bf a_n=a+(n-1)d\\\\\to \rm a_n=2+(n-1)6\\\\\to \rm a_n=4+6n-6\\\\\to \bf a_n=6n-2$