Find the sum of the arithmetic series....5 stars best answer

1) 6+10+14+...+86

2) 3+5+7+..+227

6 + 10 + 14 + ... + 86

86 = 6 + 4x
80 = 4x
20 = x

6 + 6 + 4 + 6 + 2 * 4 + ... + 6 + 20 * 4 =>
6 * 21 + 4 * (1 + 2 + 3 + ... + 20) =>
126 + 4 * (1/2) * (20) * (21) =>
126 + 2 * 20 * 21 =>
126 + 40 * 21 =>
126 + 840 =>
966


3 + 5 + 7 + ... + 227
1 + 2 * 1 + 1 + 2 * 2 + 1 + 2 * 3 + .... + 1 * 2 * 113 =>
1 * 113 + 2 * (1 + 2 + 3 + ... + 113) =>
113 + 2 * (1/2) * (113 * 114) =>
113 + 113 * 114 =>
113 * (1 + 114) =>
113 * 115 =>
12995

The formula for the sum of an arithmetic series is S = n((a1+an)/2)

S=sum
n=number of numbers in the series
a1=first value
an=last value

However, you're currently lacking n in the equation. The formula to find this is an = a1+d(n-1) where d is the rate of change between each number (in these cases, 4 and 2 respectively).

I'll do #1 for you:

an = a1 + d(n-1)

86 = 6 + 4(n-1)
80 = 4(n-1) ---Subtracted 6 from both sides
80 = 4n-4 ------Distributed
84 = 4n --------Added 4 to both sides
21 = n --------Divided by 4

S = n((a1+an)/2)
S = 21((6+86)/2)
S = 21(92/2)
S = 21(46)
S = 966

Thus, the sum is 966.

So to find the sum of an arithmetic series we need first term last term, and the number of terms
we dont seem to have the number of terms. And it seems counting is quite inefficient

To find the number of terms we subtract the highest term from the lowest then divide by the common difference.

86- 6
80
Then we take the floor function( meaning round down) of 80 / 4 which is
20 therefore there are 20 terms
so we go to our formula

number of terms(first term + last term) all divided by 2

20(6+86) divided by 2
10(6+86)
60 + 860
= 940

Now try number two :D

1.) 86 is the 21st term because to find n you use the equation an=(n-1)d+a1

an is the value of the term of the "nth" term (the one you're trying to find)
n is the number of the term in the sequence (1st, 2nd, 3rd...)
d is the amount of change between each number
a1 is just the first term

an=86
n=?
d=4
a1=6, so
86=(n-1)4+6
86=4n-4+6
86=4n+2
4n=84
n=21

From there, you use this equation: Sn=n/2(a1+an) where a1 is the value of the first number (6) and an is the value of the nth term (86)

Sn=21/2(6+86)
Sn=10.5(92)
Sn=966

2) Do the same thing with different numbers; I got 12995, but I might have messed up.