Question 1

What is a sequence in mathematics?

Accepted Answer

A sequence is an ordered list of numbers following a specific pattern or rule. Each number in the sequence is called a term. Sequences can be finite (having a specific number of terms) or infinite (continuing forever). Common types include arithmetic sequences (constant difference between terms) and geometric sequences (constant ratio between terms).

Question 2

How do you find the nth term of a sequence?

Accepted Answer

To find the nth term: For arithmetic sequences, use aₙ = a₁ + (n-1)d, where a₁ is the first term and d is the common difference. For geometric sequences, use aₙ = a₁ × r^(n-1), where r is the common ratio. For example, in the arithmetic sequence 2, 5, 8, 11..., the 10th term is a₁₀ = 2 + (10-1)(3) = 29.

Question 3

What is an arithmetic sequence?

Accepted Answer

An arithmetic sequence (or arithmetic progression) is a sequence where the difference between consecutive terms is constant. This difference is called the common difference (d). For example, 3, 7, 11, 15, 19... is an arithmetic sequence with d = 4. The general formula is aₙ = a₁ + (n-1)d.

Question 4

What is a geometric sequence?

Accepted Answer

A geometric sequence (or geometric progression) is a sequence where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio (r). For example, 2, 6, 18, 54... is a geometric sequence with r = 3. The general formula is aₙ = a₁ × r^(n-1).

Question 5

How do you calculate the sum of an arithmetic sequence?

Accepted Answer

The sum of the first n terms of an arithmetic sequence is given by: Sₙ = n/2 × (2a₁ + (n-1)d) or Sₙ = n/2 × (first term + last term). For example, the sum of the first 10 terms of 2, 5, 8, 11... is S₁₀ = 10/2 × (2(2) + (10-1)(3)) = 5 × 31 = 155.

Question 6

How do you find the sum of a geometric sequence?

Accepted Answer

The sum of the first n terms of a geometric sequence is: Sₙ = a₁(1 - r^n)/(1 - r) when r ≠ 1. For example, for the sequence 2, 6, 18, 54... (where a₁ = 2, r = 3), the sum of the first 5 terms is S₅ = 2(1 - 3⁵)/(1 - 3) = 2(-242)/(-2) = 242.

Question 7

What is the common difference in an arithmetic sequence?

Accepted Answer

The common difference (d) is the constant value added to each term to get the next term in an arithmetic sequence. To find it, subtract any term from the next term: d = aₙ - aₙ₋₁. For example, in the sequence 5, 8, 11, 14..., the common difference is d = 8 - 5 = 3.

Question 8

What is the common ratio in a geometric sequence?

Accepted Answer

The common ratio (r) is the constant factor by which each term is multiplied to get the next term in a geometric sequence. To find it, divide any term by the previous term: r = aₙ/aₙ₋₁. For example, in the sequence 3, 6, 12, 24..., the common ratio is r = 6/3 = 2.

Question 9

What is a Fibonacci sequence?

Accepted Answer

The Fibonacci sequence is a special sequence where each term is the sum of the two preceding terms: F(n) = F(n-1) + F(n-2), starting with F(0) = 0 and F(1) = 1. The sequence begins: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34... It appears frequently in nature and has many mathematical applications.

Question 10

How are sequences used in real life?

Accepted Answer

Sequences have many practical applications: arithmetic sequences in financial planning (regular deposits), geometric sequences in population growth and compound interest, Fibonacci sequences in nature (plant spirals, shell patterns), computer algorithms, physics (wave patterns), and economics (inflation calculations). They help model and predict patterns in various fields.

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