Solve the equation ` -4+(-1)+2+ … +x =437.`

Explanation

` :' ` Given equation is, ` -4+(-1)+2+ … +x =437 " "` …(i)
Here, ` -4-1+2+ … +x ` form an AP with first term `= -4` common difference `=3`
`a_(n)=l=x`
` :' n`th term of an AP, `a_(n)=l=a+(n-1)d`
` implies x= -4+(n-1)3`
`implies (x+4)/(3)=n-1impliesn=(x+7)/(3) " "` ...(ii)
` :. ` Sum of an AP, `S_(n)=(n)/(2)[2a+(n-1)d]`
`S_(n)=(x+7)/(2xx3)[2(-4)+((x+4)/(3))*3]`
` " "(x+7)/(2xx3)(-8+x+4)=((x+7)(x-4))/(2xx3)`
From Eq. (i),
`S_(n)=437`
`((x+7)(x-4))/(2xx3)=437`
`impliesx^(2)+7x-4x-28=874xx3`
`implies x^(2) +3x-2650=0`
`x=(-3+-sqrt((3)^(2)-4(-2650)))/(2) " " ` [by quardratic formula]
`x=(-3+-sqrt(9+10600))/(2)`
`x=(-3+-sqrt(10609))/(2)=(-3+-103)/(2)=(100)/(2),(-106)/(2)`
`x=50, -53`
Here, x cannot be negative. i.e., `x ne -53`
also, for `x= -53`, n will be negatuve which is not possible
Hence, the required value of x is 50.