Question

What is the sum of the sequence `5, 10, 15, 20, 25, 30, 35, 40, 45, 50`?

Answer 1

275

Explanation:

This is an`color(blue)" Arithmetic sequence "`

The standard sequence being :

a , a+d , a+2d , a+3d , ............... , a+(n-1)d

where a , is the 1st term and d , the common difference.

The sum to n terms of the sequence is :

`color(red)(|bar(ul(color(white)(a/a)color(black)( S_n = n/2 [ 2a + (n - 1)d])color(white)(a/a)|)))`

here a = 5 , d = 10-5 = 15-10 = 5 and n = 10

`rArr S_(10) = 10/2[ 2xx5 + 9xx5 ] = 5 xx55 = 275`

Answer 2

`275`

Explanation:

Notice that if we match up the first and last pairs in the sequence, we always have a pair that adds to `55`:

`color(red)5,color(blue)10,color(green)15,color(orange)20,color(brown)25,color(brown)30,color(orange)35,color(green)40,color(blue)45,color(red)50`

We see that:

`color(red)5+color(red)50=ul55`

`color(blue)10+color(blue)45=ul55`

`color(green)15+color(green)40=ul55`

`color(orange)20+color(orange)35=ul55`

`color(brown)25+color(brown)30=ul55`

Thus, the sum of the entire sequence is just `5` groupings of `55`, which is `5xx55=275`.