Question:
Find the average of odd numbers from 15 to 1709
Correct Answer
862
Solution And Explanation
Solution
Method (1) to find the average of the odd numbers from 15 to 1709
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 15 to 1709 are
15, 17, 19, . . . . 1709
After observing the above list of the odd numbers from 15 to 1709 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 15 to 1709 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 15 to 1709
The First Term (a) = 15
The Common Difference (d) = 2
And the last term (ℓ) = 1709
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 1709
= 15 + 1709/2
= 1724/2 = 862
Thus, the average of the odd numbers from 15 to 1709 = 862 Answer
Method (2) to find the average of the odd numbers from 15 to 1709
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 15 to 1709 are
15, 17, 19, . . . . 1709
The odd numbers from 15 to 1709 form an Arithmetic Series in which
The First Term (a) = 15
The Common Difference (d) = 2
And the last term (ℓ) = 1709
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 1709
1709 = 15 + (n – 1) × 2
⇒ 1709 = 15 + 2 n – 2
⇒ 1709 = 15 – 2 + 2 n
⇒ 1709 = 13 + 2 n
After transposing 13 to LHS
⇒ 1709 – 13 = 2 n
⇒ 1696 = 2 n
After rearranging the above expression
⇒ 2 n = 1696
After transposing 2 to RHS
⇒ n = 1696/2
⇒ n = 848
Thus, the number of terms of odd numbers from 15 to 1709 = 848
This means 1709 is the 848th term.
Finding the sum of the given odd numbers from 15 to 1709
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 1709
= 848/2 (15 + 1709)
= 848/2 × 1724
= 848 × 1724/2
= 1461952/2 = 730976
Thus, the sum of all terms of the given odd numbers from 15 to 1709 = 730976
And, the total number of terms = 848
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 1709
= 730976/848 = 862
Thus, the average of the given odd numbers from 15 to 1709 = 862 Answer
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