It is given that if the sum of the first n terms of an Ap is 4n-n2 what is first term? what is the sum of first two terms? what is the second term? similarly find the 3rd and 10th and nth terms.
Sir please give me a detailed solution of this question as it is very important for board examination and it is from ML Aggarwal Avichal Publication Arithmetic Progression Chapter 9 Question no 17
We have,
Sn​=4n−n2
Taking n=1 and we get,
S1​=4×1−12
=4−1
S1​=3
So, the sum of fristr term of an A.P. is 3.
But, sum of first term will be the first term.
So, first term a=3.
Now, taking n=2 and we get,
Sn​=4n−n2
S2​=4×2−22
S2​=4
Then, sum of first 2 terms is 4.
Now, finding Secondterm.
Then,
According to given question,
Firstterm+Secondterm=4
3+a2​=4
a2​=1
Hence the A.P.is
First term a=3
Common difference
d=Secondterm-firstterm
d=1-3
=-2
The third term of
a3​=a+2d
a3​=3+2×(−2)
a3​=−1
Similarly, 10th term
a10​=a+9d
=3+9(−2)
=3−18
=−15
Then,
nth term
an​=a+(n−1)d
an​=3+(n−1)(−2)
an​=3−2n+2
an​=5−2n