Arithmetic Series and Sum Formulas

Vocabulary flashcards covering key concepts related to the sum of arithmetic sequences and their terms.

S_n

The sum of the first n terms of an arithmetic sequence; formula: S_n = (n/2)[2a + (n−1)d].

a

The first term of an arithmetic sequence.

d

The common difference between consecutive terms in an arithmetic sequence.

l

The last term of the first n terms; l = a + (n−1)d.

S_n = (n/2)(a + l)

Alternative formula for the sum of the first n terms using the first term a and the last term l.

Arithmetic sequence

A sequence in which the difference between consecutive terms is constant (the common difference d).

S_n

The sum of the first n terms of an arithmetic sequence; formula: S_n = (n/2)[2a + (n−1)d].

a

The first term of an arithmetic sequence.

d

The common difference between consecutive terms in an arithmetic sequence.

l

The last term of the first n terms; l = a + (n−1)d.

S_n = (n/2)(a + l)

Alternative formula for the sum of the first n terms using the first term a and the last term l.

Arithmetic sequence

A sequence in which the difference between consecutive terms is constant (the common difference d).

Problem Solving with Arithmetic Sequences

To solve problems involving arithmetic sequences:

1. Identify the known variables (e.g., a, d, n, l, S_n).
2. Determine what needs to be found.
3. Choose the appropriate formula(s) based on knowns and unknowns:
   • l = a + (n−1)d
   • S_n = (n/2)[2a + (n−1)d]
   • S_n = (n/2)(a + l)
4. Substitute known values into the chosen formula(s) and solve for the unknown variable.

S_n

The sum of the first n terms of an arithmetic sequence; formula: S_n = (n/2)[2a + (n−1)d].

a

The first term of an arithmetic sequence.

d

The common difference between consecutive terms in an arithmetic sequence.

l

The last term of the first n terms; l = a + (n−1)d.

S_n = (n/2)(a + l)

Alternative formula for the sum of the first n terms using the first term a and the last term l.

Arithmetic sequence

A sequence in which the difference between consecutive terms is constant (the common difference d).

Problem Solving with Arithmetic Sequences

To solve problems involving arithmetic sequences:

1. Identify the known variables (e.g., a, d, n, l, S_n).
2. Determine what needs to be found.
3. Choose the appropriate formula(s) based on knowns and unknowns:
   • l = a + (n−1)d
   • S_n = (n/2)[2a + (n−1)d]
   • S_n = (n/2)(a + l)
4. Substitute known values into the chosen formula(s) and solve for the unknown variable.

The nth term of an arithmetic sequence (a_n)

The formula for the nth term of an arithmetic sequence is a_n = a + (n-1)d. It allows you to find any term in the sequence given the first term, the common difference, and the term's position.

How to determine if a sequence is arithmetic?

Check if the difference between consecutive terms is constant. This constant difference is called the common difference (d).

Example of an arithmetic sequence

An example of an arithmetic sequence is 3, 7, 11, 15, …. Here, the first term a=3 and the common difference d=4.

How do you find the common difference (d) of an arithmetic sequence?

Subtract any term from its succeeding term. For example, if the terms are a1, a2, a3, …, then d = a2 - a1 or d = a3 - a_2.

When is S_n = (n/2)[2a + (n−1)d] most useful?

This formula is useful when the first term (a), the common difference (d), and the number of terms (n) are known, and you need to find the sum of the first n terms (S_n).

When is S_n = (n/2)(a + l) most useful?

This formula is most efficient when the first term (a), the last term (l), and the number of terms (n) are known, and you need to find the sum (S_n).

How to find the number of terms (n) if S_n, a and d are known?

Substitute the known values of Sn, a, and d into the formula Sn = (n/2)[2a + (n−1)d] and solve the resulting quadratic equation for n. Remember that n must be a positive integer.

Set up the appropriate sum formula (Sn = (n/2)[2a + (n−1)d] or Sn = (n/2)(a + l)) with the given values. If l is not known, use $$l = a + (n-

S_n

The sum of the first n terms of an arithmetic sequence; formula: S_n = (n/2)[2a + (n−1)d].

a

The first term of an arithmetic sequence.

d

The common difference between consecutive terms in an arithmetic sequence.

l

The last term of the first n terms; l = a + (n−1)d.

S_n = (n/2)(a + l)

Alternative formula for the sum of the first n terms using the first term a and the last term l.

Arithmetic sequence

A sequence in which the difference between consecutive terms is constant (the common difference d).

Problem Solving with Arithmetic Sequences

To solve problems involving arithmetic sequences:

1. Identify the known variables (e.g., a, d, n, l, S_n).
2. Determine what needs to be found.
3. Choose the appropriate formula(s) based on knowns and unknowns:

• l = a + (n−1)d
• S_n = (n/2)[2a + (n−1)d]
• S_n = (n/2)(a + l)

1. Substitute known values into the chosen formula(s) and solve for the unknown variable.

The nth term of an arithmetic sequence (a_n)

The formula for the nth term of an arithmetic sequence is a_n = a + (n-1)d. It allows you to find any term in the sequence given the first term, the common difference, and the term's position.

How to determine if a sequence is arithmetic?

Check if the difference between consecutive terms is constant. This constant difference is called the common difference (d).

Example of an arithmetic sequence

An example of an arithmetic sequence is 3, 7, 11, 15, …. Here, the first term a=3 and the common difference d=4.

How do you find the common difference (d) of an arithmetic sequence?

Subtract any term from its succeeding term. For example, if the terms are a1, a2, a3, …, then d = a2 - a1 or d = a3 - a_2.

When is S_n = (n/2)[2a + (n−1)d] most useful?

This formula is useful when the first term (a), the common difference (d), and the number of terms (n) are known, and you need to find the sum of the first n terms (S_n).

When is S_n = (n/2)(a + l) most useful?

This formula is most efficient when the first term (a), the last term (l), and the number of terms (n) are known, and you need to find the sum (S_n).

How to find the number of terms (n) if S_n, a and d are known?

Substitute the known values of Sn, a, and d into the formula Sn = (n/2)[2a + (n−1)d] and solve the resulting quadratic equation for n. Remember that n must be a positive integer.

How to calculate the sum (S_n) when the last term (l) is not explicitly given?

If the last term (l) is not known, but the first term (a), common difference (d), and number of terms (n) are given, use the formula Sn = (n/2)[2a + (n−1)d]. Alternatively, first calculate l using l = a + (n-1)d, then use Sn = (n/2)(a + l).

How to find the first term (a) or common difference ($

If Sn, n, and other known variables (e.g., l or one of a, d) are given, you can use the appropriate sum formula (Sn = (n/2)[2a + (n−1)d] or S_n = (n/2)(a + l)) and the formula for the last term (l = a + (n-1)d) to set up a system of equations. Solve this system to find the unknown variable (a or d).