find the sum of all od natural number from 50 to 550 with reason
Answers
Answer:To prove: Sum of ‘n’ consecutive odd numbers = n2
Step 1:
We need to understand the pattern of odd numbers sequence to prove their sum. The total of any set of sequential odd numbers beginning with 1 is always equal to the square of the number of digits, added together. If 1,3,5,7,9,11,…, (2n-1) are the odd numbers, then;
Sum of first odd number = 1
Sum of first two odd numbers = 1 + 3 = 4 (4 = 2 x 2).
Sum of first three odd numbers = 1 + 3 + 5 = 9 (9 = 3 x 3).
Sum of first four odd numbers = 1 + 3 + 5 + 7 = 16 (16 = 4 x 4).
Step 2:
The number of digits added collectively is always equal to the square root of the total number.
Sum of first odd number = 1.
The square root of 1, √1 = 1, so, only one digit was added.
Sum of consecutive two odd numbers = 1 + 3 = 4.
The square root of 4, √4 = 2, so, two digits were added.
Sum of first three consecutive odd numbers = 1 + 3 + 5 = 9.
The square root of 9, √9 = 3, so, three digits were added.
Sum of first four odd numbers = 1 + 3 + 5 + 7 = 16.
The square root of 16, √16 = 4, so, four digits were added.
Step 3:
Hence, from the above estimation, we can prove the formula to find the sum of the first n odd numbers is n x n or n2.
For example, if we put n = 21, then we have 21 x 21 = 441, which is equal to the sum of the first 21 odd numbers.
Note: If we don’t know the number of odd numbers present in a series, then the formula to determine the sum between 1 and n is (1/2(n + 1))2.
Solved Examples
Question 1: What is the sum of odd numbers from 1 to 50?
Solution: We know that, from 1 to 50, there are 25 odd numbers.
Thus, n = 25
By the formula of sum of odd numbers we know;
Sn = n2
Sn = 252 = 625
Question 2: What is the sum of odd numbers from 1 to 99?
Solution: We know that, from 1 to 99, there are 50 odd numbers.
Thus, n = 50
By the formula of sum of odd numbers we know;
Sn = 502
Sn = 502 = 2500
Step-by-step explanation:
Step-by-step explanation:
odd numbers between 0 and 50 are 1,3,5,7,9…49
Therefore, it can be observed that these odd numbers are in an A.P.
a=1,d=2,l=49
l=a+(n−1)d
49=1+(n−1)2
48=2(n−1)
n−1=24
n=25
sn=N/2(2a+n-1)d
answer = 625