# Determine the sum of the first 20 terms of an A.P. if a4 - a2 = 4 and a1 + a3 + a5 + a6 = 30.

The sum is:

Sn = (a1+ an)n/2

We need the sun of the first 20 terms, then we will substitute with n= 20

==> s20 = (a1+ a20) * 20/2

==> S20 = 10(a1+ a20)

Given:

a4-a2= 4

a1+ a3+ a5+a6= 30

We know that: if r is the common...

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The sum is:

Sn = (a1+ an)n/2

We need the sun of the first 20 terms, then we will substitute with n= 20

==> s20 = (a1+ a20) * 20/2

==> S20 = 10(a1+ a20)

Given:

a4-a2= 4

a1+ a3+ a5+a6= 30

We know that: if r is the common ratio between numbers:

a2= a1+ r

a4= a1+ 3r

a5= a1+ 4r

a6= a1+ 5r

but: a4- a2= 4

==> a1+ 3r - (a1+r) = 4

==> 2r = 4

==> r= 2

Also:

a1+ a3+ a5+ a6 = 30

==> a1+ a1+2r + a1+ 4r + a1 5r = 30

==> 4a1 + 11r = 30

==> 4a1+ 11*2 = 30

==> 4a1 + 22 = 30

==> 4a1 = 8

==> a1= 8/4= 2

==> a1= 2

Now we will calculate a20:

a20 = a1+ 19*r

= 2 + 19*2= 2 + 38 = 40

==> a20 = 40

==> S20 =10 (a1+ a20)

= 10( 2 + 40)

= 10* 42 = 420

==> S20 = 420

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