Question

Consider the counting numbers from 1 to 100
How many multiples of 7 are there below 100
Calculate the mean of multiple of 7 below 100
What is the median of multiple of 7 below 100
How many multiples are there more are median in this collection

Answer 1

Answer:

The biggest number smaller or equal to [math]y[/math] which is a multiple with [math]z[/math] is [math]\left \lfloor \frac{y}{z} \right \rfloor[/math]. In the case of [math]y=1,000[/math] and [math]z=7[/math] this number is [math]142[/math] ([math]142 \times 7=994[/math] and as you can see [math]994+7=143 \times 7=1,001 \gt 1,000[/math]).

So we have [math]\sum \limits_{a=1}^{142} 7a[/math], as the formula to work out.

Now [math]\sum \limits_{a=1}^{142} 7a=7 \times \sum \limits_{a=1}^{142} a[/math]

and [math]\sum \limits_{a=1}^{n} a=\frac{n \times (n+1)}{2}[/math]

so we have [math]7 \times \frac{142 \times 143}{2}=71,071[/math].