Question

If 1+4+7+10+........+x = 287, find the value of x.

Answer 1

Step-by-step explanation:

$1+4+7+10+....+x=287\\</p><p>Here\\</p><p>a = 1, d= 3, s_n= 287\\</p><p>\because S_n= \frac{n}{2}\{2 a+(n-1)d\}\\\\</p><p>\therefore 287= \frac{n}{2}\{2\times 1+(n-1)\times 3\}\\\\</p><p>\therefore 287\times 2= n\{2+3n-3\}\\\\</p><p>\therefore 574= n\{3n-1\}\\\\</p><p>\therefore 574= 3n^2 - n\\\\</p><p>\therefore 3n^2 - n-574=0\\\\</p><p>\therefore 3n^2 - 42n+41n-574=0\\\\</p><p>\therefore 3n(n - 14)+41(n-14)=0\\\\</p><p>(n - 14)(3n+41)=0\\\\</p><p>\therefore n-14=0\:\: or \:\: 3n+41=0\\\\</p><p>\therefore n=14\:\: or \:\: n=-\frac{41}{3} \\\\</p><p>But\:\: n \neq - \frac{41}{3}\\\\</p><p>\therefore n =14\\\\</p><p>Hence\: 14^{th}\:term \: is\: x. \\\\</p><p>\therefore a + (n-1)d = x\\\\</p><p>\therefore 1 + (14-1)\times 3= x\\\\\therefore 1 + 13\times 3= x\\\\</p><p>\therefore 1 + 39= x\\\\</p><p>\therefore 40= x\\\\</p><p>\huge \purple {\fbox {\therefore x= 40}}$