Question

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The arithmetic mean of series 5,10,15,...., 100 is​

Answer 1

Given:

• The series 5,10,15,..,100

To Find:

• The arithmetic mean of the given A.P

Solution:

The series 5,10,15,..100 is in the sum of an AP.

In the above-mentioned series,

⇒ The first term is 5 i.e., a = 5

⇒ The common difference(second term - first term) is 5 i.e., d = 5

⇒ The last term is 100

We need to find the value of "n" by using the formula,

⇒ nth term = a+(n-1)d → {equation 1}

On substituting the values in equation1 we get,

⇒ 100 = 5+(n-1)5

⇒100 = 5+5n-5

⇒ n = 100/5 = 20

The formula to find the sum of an AP is given by,

⇒ $S_n$  = (n/2)[2a + (n-1)d] → {equation 2}

On substituting the values in equation 2 we get,

⇒ $S_n$ = (20/2)[2×5+(20-1)×5] {solving for the terms in bracket}

⇒ $S_n$ = 10[10+(19×5)] = 10[10+95] {adding and multiplying the terms}

⇒ $S_n$ = 10(105) = 1050 {multiplying the terms}

To find the arithmetic mean we use the formula given by,

Arithmetic mean = sum of the terms/number of the terms

⇒ Arithmetic mean = 1050/20 = 52.5 {dividing the terms}

∴ The arithmetic mean of the series 5,10,15,..,100 = 52.5.

Answer 2

Given : The series is 5,10,15,....,100

To find : The arithmetic mean of the given series.

Solution :

The arithmetic mean is 52.5

We can simply solve this mathematical problem by using the following mathematical process. (our goal is to calculate the arithmetic mean)

Now, the consecutive two numbers of the given series have equal difference. So, the given series is an AP series.

According to the AP series formulas :

nth term of the series = a + (n-1) × d

Sum of the terms = n/2 [2a + (n-1) × d]

Here,

First term of AP (a) = 5

Common difference (d) = Second term - First term = 10-5 = 5

Number of terms (n) = ?

So, we have to calculate the value of n.

According to the first formula,

100 = 5 + (n-1) × 5

100 = 5 + 5n - 5

5n = 100

n = 20

So, the sum of the terms will be :

= 20/2 × [(2×5) + (20-1) × 5]

= 10 × (10+95)

= 10 × 105

= 1050

Now, arithmetic mean will be :

= Sum of the terms ÷ Number of terms

= 1050 ÷ 20

= 52.5

(This will be considered as the final result.)

Hence, the arithmetic mean will be 52.5