USE SUMMATION NOTATION TO WRITE A ARITHMETIC SERIES FOR THESE TERMS. PLEASE SHOW HOW TO DO IT.

15+25+35+...; N=7
4+8+12+...; N=4
3+7+11+...;N=8

I will do the first one.

look at 15, 25, 35, ..
there is an increase of 10 from one term to the next
so the formula will be (10n +/- something)

so if n= 1, most sigma notations start with n=1,
our first term is 10(1) + ?
what do I have to add to make the first term 15?
Isn't it 5?
so we have the expression (10n + 5)

test it: do I get 35 when n = 3 ?? YEAAH

Sure, here's how you can use summation notation to write an arithmetic series for each of these terms:

1) For the series 15 + 25 + 35 + ... with N = 7:
The first term is 15, and the common difference between consecutive terms is 10.
To write it in summation notation, we can use the formula for the nth term of an arithmetic series: an = a1 + (n-1)d.
So for this series, we have:
∑(k=1 to 7) (15 + (k-1)10)

2) For the series 4 + 8 + 12 + ... with N = 4:
The first term is 4, and the common difference between consecutive terms is 4.
Using the nth term formula, we can write it in summation notation as:
∑(k=1 to 4) (4 + (k-1)4)

3) For the series 3 + 7 + 11 + ... with N = 8:
The first term is 3, and the common difference between consecutive terms is 4.
Using the nth term formula, we can write it in summation notation as:
∑(k=1 to 8) (3 + (k-1)4)

Hope that helps!

Sure! To write an arithmetic series using summation notation, we need to find the formula for the general term of the series.

The general term of an arithmetic series is given by:
a_n = a_1 + (n - 1)d

Where:
a_n is the n-th term of the series,
a_1 is the first term of the series, and
d is the common difference between consecutive terms.

1. For the series 15+25+35+... and N = 7:
In this series, the first term (a_1) is 15, and the common difference (d) is 10 (obtained by subtracting consecutive terms). We can plug these values into the formula:
a_n = 15 + (n - 1)(10)

Therefore, using summation notation, we can write the series as:
∑(n=1 to 7) (15 + (n - 1)(10))

2. For the series 4+8+12+... and N = 4:
In this series, the first term (a_1) is 4, and the common difference (d) is 4 (obtained by subtracting consecutive terms). We can plug these values into the formula:
a_n = 4 + (n - 1)(4)

Therefore, using summation notation, we can write the series as:
∑(n=1 to 4) (4 + (n - 1)(4))

3. For the series 3+7+11+... and N = 8:
In this series, the first term (a_1) is 3, and the common difference (d) is 4 (obtained by subtracting consecutive terms). We can plug these values into the formula:
a_n = 3 + (n - 1)(4)

Therefore, using summation notation, we can write the series as:
∑(n=1 to 8) (3 + (n - 1)(4))

To express arithmetic series using summation notation, we need to identify the first term, the common difference, and the number of terms.

For the series 15+25+35+... with N=7:
First, let's find the common difference. We can observe that each term is obtained by adding 10 to the previous term. Therefore, the common difference (d) is 10.

Given that the series starts with 15, we can deduce that the first term (a₁) is 15.

The number of terms in this series is given as N=7.

Now, let's write the series using summation notation:
The general form of an arithmetic series is: Sₙ = n/2 * (a₁ + aₙ)

In this case, n represents the number of terms, Sₙ is the sum of the series, a₁ is the first term, and aₙ is the last term.

To find aₙ, we use the formula for the nth term: aₙ = a₁ + (n-1)d

Substituting the values we have:
aₙ = 15 + (7-1) * 10 = 15 + 6 * 10 = 15 + 60 = 75

Now we can write the series using summation notation:
Sₙ = 7/2 * (15 + 75)

Simplifying further,
Sₙ = 7/2 * 90
Sₙ = 7 * 45
Sₙ = 315

Hence, the summation notation for the series 15+25+35+... with N=7 is Σ(i=1 to 7) 15 + 10(i-1).

Let's apply the same process to the other two series.

For the series 4+8+12+... with N=4:
The common difference (d) is 4. The first term (a₁) is 4. The number of terms (n) is given as N=4.

Using the formulas as explained earlier:
aₙ = 4 + (4-1) * 4 = 4 + 3 * 4 = 4 + 12 = 16

The summation notation for this series is: Σ(i=1 to 4) 4 + 4(i-1).

For the series 3+7+11+... with N=8:
The common difference (d) is 4. The first term (a₁) is 3. The number of terms (n) is given as N=8.

Using the formulas:
aₙ = 3 + (8-1) * 4 = 3 + 7 * 4 = 3 + 28 = 31

The summation notation for this series is: Σ(i=1 to 8) 3 + 4(i-1).

Please note: In summation notation, the index starts from 1. The value inside the summation (e.g., 15 + 10(i-1)) represents the general term of the arithmetic series.