The number of even numbers between 260 and 340 is equal to the number of odd numbers between 260 and 340.Is it true , Prove
In an Arithmetic Progression:
an = a1 + ( n - 1 ) d
where
a1 = the initial term
d = the common difference of successive members is d,
an = then the nth term
The even numbers on this sequence are members of the arithmetic progression:
a1 = 260 , an = 340 , d = 2
an = a1 + ( n - 1 ) d
340 = 260 + ( n - 1 ) ∙ 2
340 = 260 + 2 n - 2
340 = 258 + 2 n
Subtract 258 to both sides
82 = 2 n
Divide both sides by 2
41 = n
n = 41
The odd numbers on this sequence are members of the arithmetic progression:
a1 = 261 , an = 339 , d = 2
an = a1 + ( n - 1 ) d
339 = 261 + ( n - 1 ) ∙ 2
339 = 261 + 2 n - 2
339 = 259 + 2 n
Subtract 259 to both sides
80 = 2 n
Divide both sides by 2
40 = n
n = 40
There are 41 even numbers and 40 odd numbers in that sequence.
If you include both end values, there cannot be the same number of evens and odds. There is an odd number of integers in the list, which cannot be evenly divided by 2.
To determine whether the number of even numbers between 260 and 340 is equal to the number of odd numbers between 260 and 340, we need to count the number of even and odd numbers separately.
First, let's find the number of even numbers between 260 and 340:
To determine the number of even numbers, we need to check if the last digit of each number falls into the set {0, 2, 4, 6, 8}.
The first even number in this range is 260, and the last even number is 340. The interval between consecutive even numbers is 2 (gaps of 2 between even numbers), as every even number adds 2 to the previous even number.
So, we can list the even numbers between 260 and 340:
260, 262, 264, 266, 268, 270, 272, 274, 276, 278, 280, 282, 284, 286, 288, 290, 292, 294, 296, 298, 300, 302, 304, 306, 308, 310, 312, 314, 316, 318, 320, 322, 324, 326, 328, 330, 332, 334, 336, 338, 340.
Counting these numbers, we find that there are 41 even numbers between 260 and 340.
Next, let's find the number of odd numbers between 260 and 340:
To determine the number of odd numbers, we need to check if the last digit of each number falls into the set {1, 3, 5, 7, 9}.
The first odd number in this range is 261, and the last odd number is 339. The interval between consecutive odd numbers is 2 (gaps of 2 between odd numbers), as every odd number adds 2 to the previous odd number.
So, we can list the odd numbers between 260 and 340:
261, 263, 265, 267, 269, 271, 273, 275, 277, 279, 281, 283, 285, 287, 289, 291, 293, 295, 297, 299, 301, 303, 305, 307, 309, 311, 313, 315, 317, 319, 321, 323, 325, 327, 329, 331, 333, 335, 337, 339.
Counting these numbers, we find that there are 40 odd numbers between 260 and 340.
Since the number of even numbers (41) is not equal to the number of odd numbers (40) between 260 and 340, it is not true that the number of even numbers between 260 and 340 is equal to the number of odd numbers between 260 and 340.
To determine if the statement is true, we need to count the number of even numbers and the number of odd numbers between 260 and 340.
First, let's find the number of even numbers:
An even number is divisible by 2, so we need to check if the range of numbers between 260 and 340 is divisible by 2.
The divisible-by-2 test:
260 divided by 2 equals 130 (no remainder)
340 divided by 2 equals 170 (no remainder)
To find the number of even numbers, we subtract the smaller even number from the larger even number and divide it by 2:
(340 - 260) / 2 = 80 / 2 = 40
So, there are 40 even numbers between 260 and 340.
Now, let's find the number of odd numbers:
An odd number is not divisible by 2, so we need to check if the range of numbers between 260 and 340 is not divisible by 2.
The not-divisible-by-2 test:
260 divided by 2 equals 130 (no remainder)
340 divided by 2 equals 170 (no remainder)
To find the number of odd numbers, we subtract the smaller odd number from the larger odd number and divide it by 2:
(341 - 259) / 2 = 82 / 2 = 41
So, there are 41 odd numbers between 260 and 340.
Since the number of even numbers (40) is not equal to the number of odd numbers (41), we can conclude that the statement is false.