Question

Can someone solve this? IGCSE MATH, SEQUENCES QUESTION Book answer= (n-8)

1+2+3+4+.......+`x`= `((n-8)(n-7))/2`

Write `x` in terms of `n`

Answer 1

`x=n-8`

Explanation:

Sum of the series `1+2+3+4+........+x` is `(x(x+1))/2`

and as sum is given as `((n-8)(n-7))/2`

comparing the two we observe that while denominator is same,

in numerator, we have two numbers `x` and `x+1` former being smaller, whose difference is `1`

and same we have in `n-8` and `n-7`

Hence `x=n-8`.

Answer 2

Please see the explanation below

Explanation:

We know that the sum of natural numbers is

`sum_(k=1)^xk=x/2(x+1)`

But it is given that

`sum_(k=1)^xk=((n-8)(n-7))/2`

Therefore,

`x/2(x+1)=((n-8)(n-7))/2`

For this equation to be true, there is only one solution

`x=n-8`

as

`x+1=n-7`, `=>`, `x=n-8`