The sum of all the even numbers less than and inclusive of E is 197580.
(a) What is the value of E?
(b) What is the sum of all the odd numbers less than E?
assuming you mean positive even numbers, there are 98790 of them. So
S = 98790/2 (2*2 + 98789*2)
do the same, figuring how many odds there are, then summing them up starting with a=1, d=2
how do you get 98790 positive numbers?
how many even numbers are there up to 20?
20/2, since they are 2 apart. Count 'em up:
2,4,6,8,10,12,14,16,18,20
If you are going up to n, there are
n numbers
n/2 even numbers (multiples of 2)
n/3 multiples of 3
and so on
How do you find the value of E then?
To find the value of E and the sum of all the odd numbers less than E, we can use mathematical calculations and reasoning.
(a) Finding the value of E:
The sum of all even numbers less than or equal to E can be calculated using the arithmetic progression formula:
Sum = (n/2)(first term + last term)
Since we want the sum of even numbers, the first term of the arithmetic sequence is 2, and the common difference is 2. The last term can be represented as E if E is an even number, or E-1 if E is an odd number.
So, we can set up the equation:
197580 = (E/2)(2 + E) (Substituting last term as E if E is even, or E-1 if E is odd)
Now, we can solve this quadratic equation to find the value of E.
197580 = E^2/2 + E/2
Multiplying both sides by 2:
395160 = E^2 + E
Rearranging the equation:
E^2 + E - 395160 = 0
To solve this quadratic equation, we can use the quadratic formula:
E = (-b ± √(b^2 - 4ac))/(2a)
Using a = 1, b = 1, and c = -395160:
E = (-1 ± √(1^2 - 4(1)(-395160)))/(2(1))
Simplifying the equation:
E = (-1 ± √(1 + 1580640))/2
E = (-1 ± √(1580641))/2
Calculating the square root:
E ≈ (-1 ± 1257.989)/2
Since we are looking for a positive value of E, we can ignore the negative solution:
E ≈ (1257.989)/2
E ≈ 628.995
Therefore, the value of E is approximately 629.
(b) Finding the sum of all odd numbers less than E:
To find the sum of odd numbers less than a given number, we can use the formula for the sum of an arithmetic series:
Sum = (n/2)(first term + last term)
In this case, since we want the sum of odd numbers, the first term of the arithmetic sequence is 1 and the common difference is 2. The last term is E-1 if E is even, or E-2 if E is odd.
So, the sum of all odd numbers less than E can be calculated as:
Sum = ((E-1)/2)(1 + (E-2))
Substituting the value of E as 629:
Sum = (628/2)(1 + (629-2))
Sum = 314(1 + 627)
Sum = 314(628)
Sum ≈ 197,192
Therefore, the sum of all odd numbers less than E is approximately 197,192.