Q. 113.9( 249 Votes )

# If the sum of the first n terms of an AP is 4n - n^{2}, what is the first term (that is S_{1} )?

What is the sum of first two terms? What is the second term? Similarly, find the 3rd, the 10th and the nth terms.

Answer :

As the sum till first term will only contain first term,

S_{1} = a_{1}

We can find the first term as follows:

S_{n} = 4n - n^{2}

So, by putting the values in place of 1 we will get the sum till that term

S_{1} = 4 × 1 - 1^{2} = 3

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

S_{2} = 4 × 2 - 2^{2}

= 8 - 4 = 4

Hence, S_{2}= 4

Now we know by the formula that: S

_{n}- S

_{n - 1}= a

_{n}

Hence; second term = 4 – 3 = 1

And second term - first term = common difference of APTherefore, common difference, d = 1 – 3 = - 2

Now we have the AP as followsFirst 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,

a

_{n}= a + (n - 1)d

Putting the value of a and d in above formula we get

a

_{n}= 3 + (n - 1) -2

a

_{n}= 3 - 2n + 2

a

_{n}= 5 - 2n

So, 3rd term can be can be calculated by just replacing n with 3

a_{3} = 5 - 2 x 3

a_{3} = -1

Similarly

a_{10} = 5 - 2x 10

a_{10} = -15

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

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