How many times the digit 5 represent from 1 to 1000?

Odd numbers in general can be defined in many ways. One way to understand odd numbers is any number that is not a multiple of 2 is known as an odd number. Odd numbers 1 to 1000 are a set of all the non-multiples of 2 that lie between 1 to 1000 such as 11, 103, 999, and so on. Odd numbers 1 to 1000 can also be identified as all the numbers in this range ending with the odd digits such as 1, 3, 5, 7, and 9.

List of Odd Numbers from 1 to 1000

We will be listing down all the odd numbers from 1 to 1000 in this section. There are a total of 500 odd numbers from 1 to 1000. We know that odd numbers always end with an odd digit such as 1, 3, 5, 7, and 9. Therefore, the smallest odd number in this range of 1 to 1000 is 1 and the largest odd number is 999. The algorithm used to list down the odd numbers is adding 2 to the previous odd number. For instance, 1 is the first odd number, followed by 1 + 2 = 3, 3 + 2 = 5, and so on. Based on this logic let us now look into the list of all the odd numbers from 1 to 1000 as shown below.

1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49,

51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99,

101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 121, 123, 125, 127, 129, 131, 133, 135, 137, 139, 141, 143, 145, 147, 149,

151, 153, 155, 157, 159, 161, 163, 165, 167, 169, 171, 173, 175, 177, 179, 181, 183, 185, 187, 189, 191, 193, 195, 197, 199,

201, 203, 205, 207, 209, 211, 213, 215, 217, 219, 221, 223, 225, 227, 229, 231, 233, 235, 237, 239, 241, 243, 245, 247, 249,

251, 253, 255, 257, 259, 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, 341, 343, 345, 347, 349,

351, 353, 355, 357, 359, 361, 363, 365, 367, 369, 371, 373, 375, 377, 379, 381, 383, 385, 387, 389, 391, 393, 395, 397, 399,

401, 403, 405, 407, 409, 411, 413, 415, 417, 419, 421, 423, 425, 427, 429, 431, 433, 435, 437, 439, 441, 443, 445, 447, 449,

451, 453, 455, 457, 459, 461, 463, 465, 467, 469, 471, 473, 475, 477, 479, 481, 483, 485, 487, 489, 491, 493, 495, 497, 499,

501, 503, 505, 507, 509, 511, 513, 515, 517, 519, 521, 523, 525, 527, 529, 531, 533, 535, 537, 539, 541, 543, 545, 547, 549,

551, 553, 555, 557, 559, 561, 563, 565, 567, 569, 571, 573, 575, 577, 579, 581, 583, 585, 587, 589, 591, 593, 595, 597, 599,

601, 603, 605, 607, 609, 611, 613, 615, 617, 619, 621, 623, 625, 627, 629, 631, 633, 635, 637, 639, 641, 643, 645, 647, 649,

651, 653, 655, 657, 659, 661, 663, 665, 667, 669, 671, 673, 675, 677, 679, 681, 683, 685, 687, 689, 691, 693, 695, 697, 699,

701, 703, 705, 707, 709, 711, 713, 715, 717, 719, 721, 723, 725, 727, 729, 731, 733, 735, 737, 739, 741, 743, 745, 747, 749,

751, 753, 755, 757, 759, 761, 763, 765, 767, 769, 771, 773, 775, 777, 779, 781, 783, 785, 787, 789, 791, 793, 795, 797, 799,

801, 803, 805, 807, 809, 811, 813, 815, 817, 819, 821, 823, 825, 827, 829, 831, 833, 835, 837, 839, 841, 843, 845, 847, 849,

851, 853, 855, 857, 859, 861, 863, 865, 867, 869, 871, 873, 875, 877, 879, 881, 883, 885, 887, 889, 891, 893, 895, 897, 899,

901, 903, 905, 907, 909, 911, 913, 915, 917, 919, 921, 923, 925, 927, 929, 931, 933, 935, 937, 939, 941, 943, 945, 947, 949,

951, 953, 955, 957, 959, 961, 963, 965, 967, 969, 971, 973, 975, 977, 979, 981, 983, 985, 987, 989, 991, 993, 995, 997, 999.

Sum of Odd Numbers 1 to 1000

We will be finding the sum of odd numbers 1 to 1000 using the sum of odd numbers formula. According to the sum of odd numbers formula, the sum of first n odd numbers is given by n2 where n is a natural number and represents the number of terms. Thus, the sum of first n odd numbers will be represented as 1 + 3 + 5 +...+ n terms = n2.

We know that the smallest and the largest odd number in the range of 1 to 1000 will be 1 and 999. Thus, the first term is 1 and the last term is 999. The number of odd numbers between 1 to 1000 is 500, hence the number of terms n = 500.

By using the sum of first n odd numbers formula, and substituting the value of n = 500, the sum of odd numbers 1 to 1000 will be calculated as follows:

Sum = 1 + 3 + ... + 999 = n2
Sum = 5002 = 250000

Therefore the sum of odd numbers 1 to 1000 is 250000.

Related Articles

Check these articles related to the concept of odd numbers 1 to 1000.

• Odd Numbers
• Odd Numbers 1 to 100
• Even and Odd Numbers
• Whole Numbers

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Odd numbers are consecutively written $1,3,5,7,9,...,999999$. How many times does digit $5$ appear?
I have attempted to form the following strings by adding $0s$:
$000001$
$000003$
$000005$
$...$
$999999$
But can't go any further.

asked Oct 23, 2016 at 9:04

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For given $r\geq1$ consider the list of all odd natural numbers between $0$ and $10^r$.

There are $n:={1\over2}\cdot 10^r$ numbers in this list. At each of the first $r-1$ decimal places ${1\over10}$ of all $n$ numbers have the digit $5$, and at the last decimal place ${1\over5}$ of all $n$ numbers have the digit $5$. The total number $N_r$ of appearances of a $5$ in this list is therefore given by $$N_r=(r-1)\cdot{n\over10}+{n\over5}={(r+1)n\over10}={r+1\over2}\cdot10^{r-1}\ .$$ In particular $N_6=350\,000$.

answered Oct 23, 2016 at 15:00

Christian BlatterChristian Blatter

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HINT: the digit $5$ appears 1) as the last digit on the right, one time for each progressive sequence of $10$ units; 2) as the second to last digit on the right, $10$ times for each progressive sequence of $100$ units; 3) as the third to last digit on the right, $100$ times for each progressive sequence of $1000$ units; and so on.

answered Oct 23, 2016 at 9:14

AnatolyAnatoly

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Totally you will get 3 lakhs of 5s in all off numbers from 1 to 999999. when you consider all odd digits from 1 to 100, in one's place you will get 10 5s. So in 1 to 999999, you will have 10*1000000/100 number of 5s in one's place.simillarly consider for 10s,100s,1000s,10000s,100000s place add all these you will get 100000+50000+50000+50000+50000,which is equal to 300000

answered Oct 23, 2016 at 9:56

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First count how many numbers contain exactly one 5. The 5 can be in any of the six digit-places, and each of the other five digit-places can be 1, 3, 7, or 9. Hence there are $6\cdot4^5$ numbers containing exactly one 5.

Now count how many contain exactly two 5s. There are $^6\mathrm C_2$ choices of digit-places for the two 5s, and each of the other four places can be 1, 3, 7, or 9. Hence there are $^6\mathrm C_2\cdot4^4$ such numbers, and 5 occurs $2\cdot^6\mathrm C_2\cdot4^4$ among them.

Continuing in this way, we find that the number of times 5 occurs is $$\sum_{r=1}^6\,r\cdot^6\mathrm C_r\cdot4^{6-r}$$

answered Oct 23, 2016 at 10:45

George LawGeorge Law

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