Question:
Find the average of odd numbers from 15 to 1697
Correct Answer
856
Solution And Explanation
Solution
Method (1) to find the average of the odd numbers from 15 to 1697
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 15 to 1697 are
15, 17, 19, . . . . 1697
After observing the above list of the odd numbers from 15 to 1697 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 15 to 1697 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 15 to 1697
The First Term (a) = 15
The Common Difference (d) = 2
And the last term (ℓ) = 1697
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 1697
= 15 + 1697/2
= 1712/2 = 856
Thus, the average of the odd numbers from 15 to 1697 = 856 Answer
Method (2) to find the average of the odd numbers from 15 to 1697
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 15 to 1697 are
15, 17, 19, . . . . 1697
The odd numbers from 15 to 1697 form an Arithmetic Series in which
The First Term (a) = 15
The Common Difference (d) = 2
And the last term (ℓ) = 1697
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 1697
1697 = 15 + (n – 1) × 2
⇒ 1697 = 15 + 2 n – 2
⇒ 1697 = 15 – 2 + 2 n
⇒ 1697 = 13 + 2 n
After transposing 13 to LHS
⇒ 1697 – 13 = 2 n
⇒ 1684 = 2 n
After rearranging the above expression
⇒ 2 n = 1684
After transposing 2 to RHS
⇒ n = 1684/2
⇒ n = 842
Thus, the number of terms of odd numbers from 15 to 1697 = 842
This means 1697 is the 842th term.
Finding the sum of the given odd numbers from 15 to 1697
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 1697
= 842/2 (15 + 1697)
= 842/2 × 1712
= 842 × 1712/2
= 1441504/2 = 720752
Thus, the sum of all terms of the given odd numbers from 15 to 1697 = 720752
And, the total number of terms = 842
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 1697
= 720752/842 = 856
Thus, the average of the given odd numbers from 15 to 1697 = 856 Answer
Similar Questions
(1) Find the average of the first 4662 even numbers.
(2) Find the average of the first 1885 odd numbers.
(3) Find the average of the first 3512 odd numbers.
(4) Find the average of even numbers from 6 to 956
(5) Find the average of odd numbers from 3 to 177
(6) What is the average of the first 1952 even numbers?
(7) What is the average of the first 13 even numbers?
(8) Find the average of odd numbers from 13 to 327
(9) Find the average of the first 4563 even numbers.
(10) Find the average of the first 761 odd numbers.