Question

6. Find the average of first 25 multiples of 5.
a) 65
b) 60
c) 75
d) None​

Answer 1

$\Large{\bf{\green{\mathfrak{\dag{\underline{\underline{Given:-}}}}}}}$

• First 25 multiples of 5.

$\Large{\bf{\orange{\mathfrak{\dag{\underline{\underline{To \: Find:-}}}}}}}$

• Average

$\Large{\bf{\red{\mathfrak{\dag{\underline{\underline{Solution:-}}}}}}}$

First,

• 25 multiples of 5

Therefore,

A.P:- 5, 10, 15, ... n (n = 25)

• where,

• a = 5

• d = 5 (Since multiples of 5)

• n = 25

Finding the sum,

We know that,

$\boxed{\pink{\sf S_n \: = \: \dfrac{n}{2} \bigg[ 2a \: + \: (n \: - \: 1) \: d \bigg]}}$

Substituting the values,

• $\sf S_{25} \: = \: \dfrac{25}{2} \bigg[ 2 \: * \: 5 \: + \: (25 \: - \: 1) \: 5 \bigg]$

• $\sf S_{25} \: = \: \dfrac{25}{2} \bigg[ 10 \: + \: (24) \: 5 \bigg]$

• $\sf S_{25} \: = \: \dfrac{25}{2} \bigg[ 10 \: + \: 120 \bigg]$

• $\sf S_{25} \: = \: \dfrac{25}{2} \bigg[ 130 \bigg]$

• $\sf S_{25} \: = \: \dfrac{25}{2} \: * \: 130$

• $\sf S_{25} \: = \: \dfrac{25}{\cancel{2}} \: * \: \cancel{130}$

• $\sf S_{25} \: = \: 25 \: * \: 65$

• $\sf S_{25} \: = \: 1625$

Therefore,

• Sum of 1st 25 multiples of 5 = 1625

Now,

We know that,

$\boxed{\pink{\sf Average \: = \: \dfrac{Sum \: of \: 1st \: 25 \: multiples \: of \: 5}{Number \: of \: multiples}}}$

• Here,

• Sum of 1st 25 multiples of 5 = 1625

• Number of multiples = 25

Substituting the values,

• $\sf Average \: = \: \dfrac{1625}{25}$

• $\sf Average \: = \: \dfrac{\cancel{1625}}{\cancel{25}}$

• $\sf Average \: = \: 65$

Therefore,

• Average of 1st 25 multiples of 5 = 65.

• Correct option is (a) 65.

Answer 2

Method 1: The average = 5(1+25)/2 = 5*13 = 65.

Method 2. Sum of 25 terms of an Ap whose first term is 5 and common difference = 5 is

S25 = (25/2)[2*5 + (25–1)*5]

= (25/2)[10+24*5]

= (25/2)[10+120]

= 25*130/2

= 1625

So average of the 25 terms is 1625/25 = 65.

Method 3: Among 25 terms, the middle term will be the 13th term from either end and it will be the average of the whole series. The 13th term is 5x13 = 65.

So the average of the 25 terms which are multiples of 5 is 65.