Q.1 Who among the following is the famous mathematician associated with finding the sum of the first 100 natural numbers?

1. Pythagoras
2. Newton
3. Gauss
4. Euclid

 

(OR)

 

Q.2 In an A.P., if 𝑎 = 1, 𝑎_{n =} 20 and S_n = 399 , then the value of n  is:

1. 19
2. 21
3. 38
4. 42

Answer : 3

Gauss is the famous mathematician who found the sum of the first 100 natural numbers.

He found the sum using the arithmetic progression in which the sum of a series can be found using the first number, the last number and the difference between the consecutive numbers. Hence, the correct option is (c) Gauss.

(OR)

Answer : 3

We are given the first term of the series, the last, i.e., the 𝑛𝑡ℎ term of the series and the sum of the 𝑛 numbers.

The first number, 𝑎 = 1

The last number, 𝑎 = 20

And the sum of 𝑛 numbers, 𝑆 = 399

We know that the 𝑛𝑡ℎ term of the series, 𝑎    = 𝑎 + (𝑛 − 1)𝑑, where 𝑛 is the total terms

in the series and 𝑑 is the difference between the consecutive terms. Putting the given values in the above equation, we get

20 = 1 + (𝑛 − 1)𝑑

⇒ (𝑛 − 1)𝑑 = 20 − 1

⇒ (𝑛 − 1)𝑑 = 19

Now, the sum of the 𝑛 terms of a series is given by the formula

S_n=  $\frac{n}{2}$(2𝑎 + (𝑛 − 1)𝑑

Putting the values of given values and the value of (𝑛 − 1)𝑑 in the above equation, we get

399 =$\frac{n}{2}$ (2 × 1 + 19)

399 =  $\frac{n}{2}$× 21

n = $\frac{399\times 2}{21}$

n = 38

Hence, the correct option is (c) 38.