Question : Find the average of odd numbers from 15 to 1281
Correct Answer 648
Solution & Explanation
Solution
Method (1) to find the average of the odd numbers from 15 to 1281
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 15 to 1281 are
15, 17, 19, . . . . 1281
After observing the above list of the odd numbers from 15 to 1281 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 15 to 1281 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 15 to 1281
The First Term (a) = 15
The Common Difference (d) = 2
And the last term (ℓ) = 1281
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 1281
= 15 + 1281/2
= 1296/2 = 648
Thus, the average of the odd numbers from 15 to 1281 = 648 Answer
Method (2) to find the average of the odd numbers from 15 to 1281
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 15 to 1281 are
15, 17, 19, . . . . 1281
The odd numbers from 15 to 1281 form an Arithmetic Series in which
The First Term (a) = 15
The Common Difference (d) = 2
And the last term (ℓ) = 1281
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 1281
1281 = 15 + (n – 1) × 2
⇒ 1281 = 15 + 2 n – 2
⇒ 1281 = 15 – 2 + 2 n
⇒ 1281 = 13 + 2 n
After transposing 13 to LHS
⇒ 1281 – 13 = 2 n
⇒ 1268 = 2 n
After rearranging the above expression
⇒ 2 n = 1268
After transposing 2 to RHS
⇒ n = 1268/2
⇒ n = 634
Thus, the number of terms of odd numbers from 15 to 1281 = 634
This means 1281 is the 634th term.
Finding the sum of the given odd numbers from 15 to 1281
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 1281
= 634/2 (15 + 1281)
= 634/2 × 1296
= 634 × 1296/2
= 821664/2 = 410832
Thus, the sum of all terms of the given odd numbers from 15 to 1281 = 410832
And, the total number of terms = 634
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 1281
= 410832/634 = 648
Thus, the average of the given odd numbers from 15 to 1281 = 648 Answer
Similar Questions
(1) Find the average of even numbers from 10 to 1014
(2) Find the average of even numbers from 12 to 522
(3) Find the average of the first 3303 odd numbers.
(4) Find the average of even numbers from 8 to 1242
(5) What will be the average of the first 4650 odd numbers?
(6) What is the average of the first 1288 even numbers?
(7) Find the average of the first 4528 even numbers.
(8) Find the average of the first 3965 even numbers.
(9) Find the average of the first 1544 odd numbers.
(10) Find the average of even numbers from 8 to 672