Consider the sequence of consecutive odd numbers starting with 1 and ending with x. If 1+3+5+•••••+x=169,find the value of x

Take odd number 1 to 7, there are total of four odd numbers 1,3,5,7, so there are 2 pairs of 1,7 and 3,5 that add up to be the same. The sum is (1+7) *2, which is 16. This method only works for even number of items (hence 4 number can make two pairs). For sum of 1,3,5,7,9, which has 5 odd numbers, the sum is (1+9)*2+ median (which is coincidentally (1+9)/2). Apparently, even items will get you a even result and, odd items will get you a odd result. Since 169 is an odd result we can conclude there are odd items in the sequence. So we use the 2nd formula (1+x)n+(1+x)/2=169, all we need to figure out is what is n. When we run sequence to 1 to 9 there are 2 pairs + 1 item, 1 to 17 there are 4 pairs + 1 item, hence n=(x-1)/4. Just like median = (1+x)/2, this is another natural property of consecutive odd number sequence starting from 1. So (1+x)*(x-1)/4 + (1+x)/2=169 and simplify it we get (1+x) *(x-1+2)/4=169, (1+x)^2 =169*4, 1+x = 13*2, x =25