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

## 1+2+3+4+.......+$x$= $\frac{\left(n - 8\right) \left(n - 7\right)}{2}$ Write $x$ in terms of $n$

Aug 8, 2018

$x = n - 8$

#### Explanation:

Sum of the series $1 + 2 + 3 + 4 + \ldots \ldots . . + x$ is $\frac{x \left(x + 1\right)}{2}$

and as sum is given as $\frac{\left(n - 8\right) \left(n - 7\right)}{2}$

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 $x = n - 8$.

Aug 8, 2018

Please see the explanation below

#### Explanation:

We know that the sum of natural numbers is

${\sum}_{k = 1}^{x} k = \frac{x}{2} \left(x + 1\right)$

But it is given that

${\sum}_{k = 1}^{x} k = \frac{\left(n - 8\right) \left(n - 7\right)}{2}$

Therefore,

$\frac{x}{2} \left(x + 1\right) = \frac{\left(n - 8\right) \left(n - 7\right)}{2}$

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

$x = n - 8$

as

$x + 1 = n - 7$, $\implies$, $x = n - 8$