# If the sum of the first n terms of an AP is 4n - n2, what is the first term (that is S1 )?What is the sum of first two terms? What is the second term? Similarly, find the 3rd, the 10th and the nth terms.

As the sum till first term will only contain first term,
S1 = a1

We can find the first term as follows:
Sn = 4n - n2

So, by putting the values in place of 1 we will get the sum till that term
S1 = 4 × 1 - 12 = 3

Now; sum of first two terms can be calculated as follows:

S2 = 4 × 2 - 22

= 8 - 4 = 4

Hence, S2 = 4
Now we know by the formula that: Sn - Sn - 1 = an

Hence; second term = 4 – 3 = 1

And second term - first term = common difference of AP

Therefore, common difference,  d = 1 – 3 = - 2

Now we have the AP as follows
First term, a = 3
Common difference = -2
So AP will look like : 3, 1, -1, -3, -5, ..........................
And, nth term of an AP is given by the formula,

an = a + (n - 1)d
Putting the value of a and d in above formula we get
an = 3 + (n - 1) -2
an = 3 - 2n + 2
an = 5 - 2n

So, 3rd term can be can be calculated by just replacing n with 3
a3 = 5 - 2 x 3
a3 = -1
Similarly
a10 = 5 - 2x 10
a10 = -15

And so on we can calculate any number of terms of this AP

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