Grades 9-10 Video Solutions 2010-Levels 9&10 Video Solutions 2010 problem6

Video Transcription

Question number 6. It is evident from the picture that 1 plus 3 plus 5 plus 7 is equal to 4 times 4. What is the value of the sum 1 plus 3 plus 5 plus all the way up to 75  plus 77. So what we have in the picture is a 4 by 4 grid where the dots are arranged at the integer coordinates and they fill up the grid exactly and are grouped by odd numbers.  So there is 1 dot here, 3, then 5, and then 7 and we see that together there is 4 in the row and 4 in the column. In each row and in each column there is a dot for a total  of 16 or 4 times 4. And if we were to continue this pattern, the next item in the sum, the next term would be 9 because we would have here 1, 2, 3, 4, 5, 6, 7, 8, and 9 dots and  we can arrange them just as previously and now have a 5 by 5 square. So here we have  25 dots and that's 5 times 5 and we have the sum 1 plus 3 plus 5 plus 7 and the last here element of the pattern had 9 dots. So on the right hand side of this equation what  we have is 5 odd numbers and here we have 5 squared. So there is a relationship here between the number of odd numbers we're summing and what that sum is. And in the case of a  4 by 4 grid, here we have 4 squared and that's given by 4 odd numbers. So to answer our question  here, 1 plus 3 plus 5 all the way up to 77 is equal to some number here raised to the second power and the question is how many odd numbers are here? So that is what we have  to square. And if we have 4 odd numbers as in the example here, we notice that we can take the  last number here, add 1 to it, that would give us an 8 and divide that by 2 and we have a 4.  If we have 5 odd numbers, we take the last odd number here, 9, add 1 to it, divide that  10 by 2 and we obtain a 5. So here the value of the question mark following the same logic  we have the number of odd numbers is equal to 77 plus 1 and then divide that by 2. So that's 78 divided by 2 which comes out to 39. So we can then say that there are exactly  39 odd numbers here in this sequence, in this sum and the value here would be 39 squared or  39 times 39 and that gives us answer B.

Video Summary

The video explains how to find the sum of a series of odd numbers, specifically from 1 to 77. It illustrates the relationship between the number of odd numbers and a perfect square using a grid representation. For example, the sum 1 + 3 + 5 + 7 equals 4². The general pattern involves counting the odd numbers, adding 1 to the last odd number, dividing by 2, and squaring the result. For the sum up to 77, there are 39 odd numbers, so the result is 39² or 1521, leading to answer B.

Keywords

sum of odd numbers

perfect square

grid representation

mathematical pattern

series calculation