Question:
Find the average of odd numbers from 15 to 693
Correct Answer
354
Solution And Explanation
Solution
Method (1) to find the average of the odd numbers from 15 to 693
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 15 to 693 are
15, 17, 19, . . . . 693
After observing the above list of the odd numbers from 15 to 693 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 15 to 693 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 15 to 693
The First Term (a) = 15
The Common Difference (d) = 2
And the last term (ℓ) = 693
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 15 to 693
= 15 + 693/2
= 708/2 = 354
Thus, the average of the odd numbers from 15 to 693 = 354 Answer
Method (2) to find the average of the odd numbers from 15 to 693
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 15 to 693 are
15, 17, 19, . . . . 693
The odd numbers from 15 to 693 form an Arithmetic Series in which
The First Term (a) = 15
The Common Difference (d) = 2
And the last term (ℓ) = 693
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 15 to 693
693 = 15 + (n – 1) × 2
⇒ 693 = 15 + 2 n – 2
⇒ 693 = 15 – 2 + 2 n
⇒ 693 = 13 + 2 n
After transposing 13 to LHS
⇒ 693 – 13 = 2 n
⇒ 680 = 2 n
After rearranging the above expression
⇒ 2 n = 680
After transposing 2 to RHS
⇒ n = 680/2
⇒ n = 340
Thus, the number of terms of odd numbers from 15 to 693 = 340
This means 693 is the 340th term.
Finding the sum of the given odd numbers from 15 to 693
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 15 to 693
= 340/2 (15 + 693)
= 340/2 × 708
= 340 × 708/2
= 240720/2 = 120360
Thus, the sum of all terms of the given odd numbers from 15 to 693 = 120360
And, the total number of terms = 340
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 15 to 693
= 120360/340 = 354
Thus, the average of the given odd numbers from 15 to 693 = 354 Answer
Similar Questions
(1) Find the average of the first 4965 even numbers.
(2) Find the average of the first 3959 even numbers.
(3) Find the average of the first 4329 even numbers.
(4) Find the average of even numbers from 4 to 750
(5) Find the average of the first 3772 odd numbers.
(6) Find the average of the first 656 odd numbers.
(7) Find the average of odd numbers from 15 to 1463
(8) Find the average of even numbers from 12 to 1480
(9) Find the average of the first 2369 even numbers.
(10) Find the average of the first 3977 even numbers.