Answer:
165150 is the sum of the multiples of 3 between 100 and 1000.
Step-by-step explanation:
We need to find the sum of multiples of 3 between 100 and 1000.
First we will find the Total number of multiples of 3 between 100 and 1000.
Let a be the first multiple and l be the last multiple of 3
100 is not the multiple of 3.
101 is not the multiple of 3.
102 is the multiple of 3.
Hence first term a = 102
Similarly.
1000 is not a multiple of 3
999 is a multiple of 3
hence last term l = 999
Also d is the common difference.
hence d = 3.
Now by using Arithmetic progression formula we get;
[tex]T_n(l) =a+(n-1)d\\ 999=102+(n-1)3\\999-102=(n-1)3\\897=(n-1)3\\\frac{897}{3}=n-1\\\\n-1=299\\n=299+1\\n=300[/tex]
Hence there are 300 multiples of 3 between 100 and 1000
Now n=300, a=102, l = 999
Hence to find the sum of all the multiples we use the Sum of n terms in AP formula;
Sum of n term [tex]S_n= \frac{n}{2}(a+l)[/tex]
[tex]S_{300}= \frac{300}{2}(102+999)\\\\S_{300}= 150(102+999)\\S_{300}= 150\times 1101\\S_{300}= 165150[/tex]
Hence,165150 is the sum of the multiples of 3 between 100 and 1000.
