Q. 1 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.

Q. 2 Check whether – 150 is a term of the AP: 11, 8, 5, 2 . . .

Maths

Class 10

633

Anaisha

Qayanat

Answer / Solution

Answer / Solution

**Answer 1 :** Given that, S_{n} = 4n − n^{2}

First term will be, a = S_{1} = 4(1) − (1)^{2} = 4 − 1 = 3

Sum of first two terms = S_{2 }= 4(2) − (2)^{2} = 8 − 4 = 4

Second term, a_{2} = S_{2} − S_{1} = 4 − 3 = 1

Common difference, d = a_{2} − a = 1 − 3 = −2

Now we find the n^{th} term is using formula

a_{n} = a + (n − 1)d = 3 + (n − 1) (−2) = 3 − 2n + 2 = **5 − 2n**

Therefore, a_{3} = 5 − 2(3) = 5 − 6 = −1

a_{10} = 5 − 2(10) = 5 − 20 = −15

Hence, the sum of first two terms is 4. The second term is 1.

The 3^{rd}, 10^{th}, and n^{th} terms are −1, −15, and 5 − 2n respectively.

**Answer 2: **

For the given, A.P. 11, 8, 5, 2, …, Here First term is **a = 11**, common difference is **d = a _{2} − a_{1} = 8 − 11 = −3**

Let −150 be the n^{th} term of this A.P. Here we know, for an A.P, a_{n} = a + (n − 1) d, substitute the value from previous steps, we get

–150 = 11 + (n – 1)(–3)

–150 = 11 – 3n + 3

–164 = –3n

n = 164/3

Clearly, **n is not an integer but a fraction**, therefore, –150 is not a term of this A.P.

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