Question : Find the average of odd numbers from 13 to 1287
Correct Answer 650
Solution & Explanation
Solution
Method (1) to find the average of the odd numbers from 13 to 1287
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 13 to 1287 are
13, 15, 17, . . . . 1287
After observing the above list of the odd numbers from 13 to 1287 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 13 to 1287 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 13 to 1287
The First Term (a) = 13
The Common Difference (d) = 2
And the last term (ℓ) = 1287
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 1287
= 13 + 1287/2
= 1300/2 = 650
Thus, the average of the odd numbers from 13 to 1287 = 650 Answer
Method (2) to find the average of the odd numbers from 13 to 1287
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 13 to 1287 are
13, 15, 17, . . . . 1287
The odd numbers from 13 to 1287 form an Arithmetic Series in which
The First Term (a) = 13
The Common Difference (d) = 2
And the last term (ℓ) = 1287
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 1287
1287 = 13 + (n – 1) × 2
⇒ 1287 = 13 + 2 n – 2
⇒ 1287 = 13 – 2 + 2 n
⇒ 1287 = 11 + 2 n
After transposing 11 to LHS
⇒ 1287 – 11 = 2 n
⇒ 1276 = 2 n
After rearranging the above expression
⇒ 2 n = 1276
After transposing 2 to RHS
⇒ n = 1276/2
⇒ n = 638
Thus, the number of terms of odd numbers from 13 to 1287 = 638
This means 1287 is the 638th term.
Finding the sum of the given odd numbers from 13 to 1287
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 1287
= 638/2 (13 + 1287)
= 638/2 × 1300
= 638 × 1300/2
= 829400/2 = 414700
Thus, the sum of all terms of the given odd numbers from 13 to 1287 = 414700
And, the total number of terms = 638
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 1287
= 414700/638 = 650
Thus, the average of the given odd numbers from 13 to 1287 = 650 Answer
Similar Questions
(1) What is the average of the first 1376 even numbers?
(2) Find the average of even numbers from 12 to 776
(3) Find the average of even numbers from 12 to 614
(4) Find the average of odd numbers from 15 to 723
(5) Find the average of odd numbers from 11 to 1237
(6) What will be the average of the first 4286 odd numbers?
(7) Find the average of even numbers from 4 to 568
(8) Find the average of odd numbers from 3 to 1349
(9) Find the average of odd numbers from 5 to 677
(10) Find the average of even numbers from 8 to 670