11, 18, 25, 32, ..., a8

we notice that we have an A.P where a1= 11 and r= 7

a1= 11

a2= 11+7 = 18

a3= 11+7*2 = 25

a4 = 11+ 7*3= 32

a5= 11 + 7*4= 39

a6 = 11+ 7*5 = 46

a7 = 11+ 7*6 = 53

a8 = 11 + 7*7 = 60

Then the sum is:

S = 11 + 18 + 25 + 32 + 39 + 46 + 53 + 60 = 284

**Then the sum = 284**

We'll apply the formula of the sum of the n terms of an arithmetic sequence:

Sn = (a1+an)*n/2

Now, we'll substitute n by 8:

S8 = (a1 + a8)*8/2

S8 = (a1 + a8)*4

We'll know the value of the first 4 terms from the 8 terms.

But we know the formula of finding any term of an arithmetic series.

an =a1 + (n-1)*d, where a1 is the first term and d is the common difference.

a8 = a1 + (8-1)*d

We could calculate d from:

a2 - a1 = d

18 - 11 = 7

d = 7

a8 = 11 + 7*7

a8 = 11 + 49

a8 = 60

S8 = (11 + 60)*4

S8 = 71*4

**S8 = 284**

**The sum of the first 8 terms of the arithmetic progression, whose first term is 11 and common difference is 7, is S8 = 284.**

To find the sum of the 1st 8 terms of 11,18,25,32...

Solution:

We check for the difference:

ar = rth term.

a2-a1 = 18 - 11 = 7.

a3-a2 = 25 - 18 = 7.

a4-a2 = 32 - 25 = 7.

We conclude that this is an arithmrtic progression with starting term a1 = 11 and the common difference d = 7 between the cosecutive terms.

So the sum of n terms is given by:

Sn = (a1+an)d/2.

In our examle,

a1 = 11.

a8 = 11+(8-1)*7 = 11+7*7 = 60.

S8 = {11+60}8/2 = 71*4 = 284.

Looking as the series, we see that 18 - 11= 25 - 18= 32- 25 =7.

So we see that it is an AP with the common difference equal to 7 and the first term equal to 11.

For an AP, the nth term is expressed as a+(n-1)*d where a is the first term and d is the common difference.

Also the sum of n terms of an AP is (n/2)*(2a+(n-1)d).

Substituting the values we have here:

(n/2)*(2a+(n-1)d)= (8/2)* ( 2*11 + 7* 7)

= 4 *( 22+49)

=4*71

= 284

**The sum of the first 8 terms of the series is 284**

It is an arithmetic progression whose common difference is 7 (difference between two consecutive terms). The formula for the sum of an A.P is SUM = n/2 [2a + (n - 1)d]. From the question, n = 8 (i.e number of terms), a = 11 ( the first term), and d = 7 ( common difference). Substituting into the equation for sum of an A.P we obtain SUM = 8/2 [2 * 11 + (8 - 1)7]. ==> SUM = 4 [22 +49] ==> SUM = 4 * 71 ==> SUM = 284