Question : Find the average of odd numbers from 13 to 113
Correct Answer 63
Solution & Explanation
Solution
Method (1) to find the average of the odd numbers from 13 to 113
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 13 to 113 are
13, 15, 17, . . . . 113
After observing the above list of the odd numbers from 13 to 113 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 13 to 113 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 13 to 113
The First Term (a) = 13
The Common Difference (d) = 2
And the last term (ℓ) = 113
The average of the numbers forming an Arithmetic Series
= The first term (a) + The last term (ℓ)/2
⇒ The average of numbers forming an Arithmetic Series = a + ℓ/2
Thus, the average of the odd numbers from 13 to 113
= 13 + 113/2
= 126/2 = 63
Thus, the average of the odd numbers from 13 to 113 = 63 Answer
Method (2) to find the average of the odd numbers from 13 to 113
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 13 to 113 are
13, 15, 17, . . . . 113
The odd numbers from 13 to 113 form an Arithmetic Series in which
The First Term (a) = 13
The Common Difference (d) = 2
And the last term (ℓ) = 113
The Average of the given numbers
= Sum of the given numbers/Total number of given numbers
Thus, to find the average of the given numbers, first, we need to find their sum and the total number of given numbers
Finding the number of terms
For an Arithmetic Series, the nth term
an = a + (n – 1) d
Where
a = First term
d = Common difference
n = number of terms
an = nth term
Thus, for the given series of the odd numbers from 13 to 113
113 = 13 + (n – 1) × 2
⇒ 113 = 13 + 2 n – 2
⇒ 113 = 13 – 2 + 2 n
⇒ 113 = 11 + 2 n
After transposing 11 to LHS
⇒ 113 – 11 = 2 n
⇒ 102 = 2 n
After rearranging the above expression
⇒ 2 n = 102
After transposing 2 to RHS
⇒ n = 102/2
⇒ n = 51
Thus, the number of terms of odd numbers from 13 to 113 = 51
This means 113 is the 51th term.
Finding the sum of the given odd numbers from 13 to 113
The sum of all terms (S) in an Arithmetic Series
= n/2 (a + ℓ)
Where, n = number of terms
a = First term
And, ℓ = Last term
Thus, the sum of all terms (S) of the given odd numbers from 13 to 113
= 51/2 (13 + 113)
= 51/2 × 126
= 51 × 126/2
= 6426/2 = 3213
Thus, the sum of all terms of the given odd numbers from 13 to 113 = 3213
And, the total number of terms = 51
Since, the average of the given numbers
= Sum of the given numbers/Total number of given numbers
Thus, the average of the given odd numbers from 13 to 113
= 3213/51 = 63
Thus, the average of the given odd numbers from 13 to 113 = 63 Answer
Similar Questions
(1) Find the average of the first 4425 even numbers.
(2) Find the average of even numbers from 8 to 298
(3) Find the average of odd numbers from 15 to 1631
(4) What will be the average of the first 4934 odd numbers?
(5) Find the average of even numbers from 12 to 34
(6) Find the average of even numbers from 12 to 506
(7) Find the average of odd numbers from 11 to 1159
(8) Find the average of the first 208 odd numbers.
(9) Find the average of odd numbers from 13 to 1313
(10) Find the average of odd numbers from 11 to 775