Question

the sum of first term of an ap is 3n2/2+5n+2 find 25th term​

Answer 1

25th term is 76

Given

Sum of n terms of an AP is given by Sₙ = 3n²/2 + 5n/2

To Find

25 th 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

$\rm S_n=\dfrac{3n^2}{2}+\dfrac{5n}{2}\\\\\to \rm S_1=\dfrac{3(1)^2}{2}+\dfrac{5(1)}{2}\\\\\to \rm S_1=\dfrac{3}{2}+\dfrac{5}{2}\\\\\to \rm S_1=\dfrac{8}{2}\\\\\to \rm S_1=4\\\\\bf \to a=4$

where "a" denotes first term

$\rm S_2=\dfrac{3(2)^2}{2}+\dfrac{5(2)}{2}\\\\\to \rm S_2=\dfrac{12}{2}+\dfrac{10}{2}\\\\\to \rm S_2=6+5\\\\\to \rm S_2=11\\\\\to \rm a_1+a_2=11\\\\\to \rm a+a+d=11\\\\\to \rm 2a+d=11\\\\\to \rm 2(4)+d=11\\\\\to \bf d=3$

Now , 25th term is ,

$\bf a_{25}\\\\\to \rm a+(25-1)d\\\\\to \rm 4+(24)3\\\\\to \rm 4+72\\\\\to \bf 76$

So , 25th term is 76