Question

find the sum of all natural numbers between 3ooand 600which are divisible by 7​

Answer 1

The first one is 301
301
, 43∗7=301
43
∗
7
=
301
.

The last one is 595
595
, 85∗7=595
85
∗
7
=
595
.

595−301=294
595
−
301
=
294


2947=42+1=43
294
7
=
42
+
1
=
43


So there are 43
43
numbers in range { 300,600
300
,
600
} that are divisible by 7
7
.

The first one label
n


The second, +7
n
+
7


And so on..

∑=042+7
∑
k
=
0
42
n
+
7
k


⟹∑=042301+∑=0427
⟹
∑
k
=
0
42
301
+
∑
k
=
0
42
7
k


⟹43(301)+7(42(43)2)
⟹
43
(
301
)
+
7
(
42
(
43
)
2
)


⟹12943+6321=19264
⟹
12943
+
6321
=
19264