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Questions and Answers
Considering an arithmetic sequence, which statement accurately relates consecutive terms, $T_{n+1}$ and $T_n$, with the common difference, $d$?
Considering an arithmetic sequence, which statement accurately relates consecutive terms, $T_{n+1}$ and $T_n$, with the common difference, $d$?
- $T_{n+1} - T_n = d^2$
- $T_{n+1} \times T_n = d$
- $T_{n+1} - T_n = d$ (correct)
- $T_{n+1} + T_n = d$
Given the sequence $\frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}, \frac{1}{6}...$, what type of sequence is formed by its reciprocals $2, 3, 4, 5, 6...$?
Given the sequence $\frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}, \frac{1}{6}...$, what type of sequence is formed by its reciprocals $2, 3, 4, 5, 6...$?
- Arithmetic sequence (correct)
- Fibonacci sequence
- Geometric sequence
- Harmonic sequence
If the $n^{th}$ term of a sequence is given by $T_n = (n+1)^2$, determine whether this sequence is arithmetic.
If the $n^{th}$ term of a sequence is given by $T_n = (n+1)^2$, determine whether this sequence is arithmetic.
- Yes, because the terms are increasing quadratically.
- Cannot be determined without more information.
- No, because the difference between consecutive terms is not constant. (correct)
- Yes, because the difference between consecutive terms is constant.
In an arithmetic sequence, if the first term is 102 and the common difference is -2, what is the 21st term?
In an arithmetic sequence, if the first term is 102 and the common difference is -2, what is the 21st term?
Given an arithmetic sequence where $T_1 = -2$ and the common difference $d = 3$, determine whether the sequence is increasing or decreasing.
Given an arithmetic sequence where $T_1 = -2$ and the common difference $d = 3$, determine whether the sequence is increasing or decreasing.
What is the 15th term of an arithmetic sequence if the $n^{th}$ term is given by $T_n = 3n - 5$?
What is the 15th term of an arithmetic sequence if the $n^{th}$ term is given by $T_n = 3n - 5$?
Given an arithmetic sequence, if 85 = 3n - 5, what is the value of n?
Given an arithmetic sequence, if 85 = 3n - 5, what is the value of n?
If the last term of an arithmetic sequence is represented by $l$, the first term by $a$, and the common difference by $d$, which formula accurately represents the last term?
If the last term of an arithmetic sequence is represented by $l$, the first term by $a$, and the common difference by $d$, which formula accurately represents the last term?
Given an arithmetic sequence with $a = -42$, $d = 3$, and $l = 21$, find the number of terms ($n$).
Given an arithmetic sequence with $a = -42$, $d = 3$, and $l = 21$, find the number of terms ($n$).
If the $n^{th}$ term of a sequence is given by $T_n= a + (n-1)d$ and the value of the term is 12, given $a = -42$ and $d = 3$, find the order of the term.
If the $n^{th}$ term of a sequence is given by $T_n= a + (n-1)d$ and the value of the term is 12, given $a = -42$ and $d = 3$, find the order of the term.
Given an arithmetic sequence in which $T_2 + T_4 = 2$, and $T_5 + T_6 + T_7 = -45$, find the value of $a$ and $d$.
Given an arithmetic sequence in which $T_2 + T_4 = 2$, and $T_5 + T_6 + T_7 = -45$, find the value of $a$ and $d$.
What is the arithmetic mean of the numbers 5, 7, 9, and 11?
What is the arithmetic mean of the numbers 5, 7, 9, and 11?
Determine the 12th term ($X_3$) when inserting 11 arithmetic means between 25 and -11.
Determine the 12th term ($X_3$) when inserting 11 arithmetic means between 25 and -11.
In the given geometric sequence 3, 6, 12, ..., what is the common ratio?
In the given geometric sequence 3, 6, 12, ..., what is the common ratio?
Given a geometric sequence defined by $T_n=2 \times 3^{n-1}$, find the second term ($T_2$).
Given a geometric sequence defined by $T_n=2 \times 3^{n-1}$, find the second term ($T_2$).
Flashcards
Arithmetic sequence
Arithmetic sequence
A sequence where the difference between consecutive terms is constant.
Common difference (d)
Common difference (d)
The constant difference between consecutive terms in an arithmetic sequence
Harmonic Sequence
Harmonic Sequence
A sequence where the reciprocals of the terms form an arithmetic sequence
General Term Formula
General Term Formula
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Arithmetic mean
Arithmetic mean
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Sum of n terms (Sₙ)
Sum of n terms (Sₙ)
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Geometric Sequence
Geometric Sequence
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Common Ratio (r)
Common Ratio (r)
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Geometric Means
Geometric Means
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Geometric Mean (g.m)
Geometric Mean (g.m)
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Study Notes
- Doctor Math study notes on algebra, arithmetic, and geometric sequences.
Arithmetic Sequence
- An arithmetic sequence is a sequence where the difference between consecutive terms is constant.
- The common difference "d" in an arithmetic sequence is found by subtracting a term from its subsequent term (Tₙ₊₁ - Tₙ).
- Given an arithmetic sequence, Tₙ₊₁ - Tₙ = constant
- Presented is an example of an arithmetic sequence: (7, 10, 13, 16, 19), where d = 3.
- Presented is an example of an arithmetic sequence: (38, 33, 28, 23, 18,...), where d = -5.
Harmonic Sequence
- A harmonic sequence is a sequence whose reciprocals form an arithmetic sequence
- The sequence (1/2, 1/3, 1/4, 1/5, 1/6) is a harmonic sequence because their reciprocals (2, 3, 4, 5, 6) form an arithmetic sequence.
Finding Terms in a Sequence
- To find a specific term (Tₙ) in an arithmetic sequence, use the formula Tₙ = a + (n - 1)d, where "a" is the first term and "n" is the term number.
- For a sequence where Tₙ = 3n - 5, it is shown that the sequence is arithmetic with d = 3.
- If given an arithmetic sequence (72, 67, 62, ...), to find the tenth term (T₁₀) the solution will be: T₁₀ = 27
Last Term
- The last term "l" of an arithmetic sequence can be found using l = a + (n - 1)d.
- If given a sequence (-42, -39, -36, ..., 21) then to calculate the number of terms we perform the following: 21 = -42 + (n-1)3 resulting in n = 22.
Arithmetic Mean (AM)
- The arithmetic mean of numbers is the sum of the numbers divided by the count of the numbers.
- The arithmetic mean of 5, 7, 9, and 11 is 8.
- Given an arithmetic mean of 13 and product of 168 will have two numbers, 14 and 12 which can replace "x" and "y" in any order.
Arithmetic Series
- A series is described as arithmetic if the sequence of its terms forms an arithmetic progression.
- The sum of "n" terms of an arithmetic series, denoted as Sₙ, can be calculated using Sₙ = n/2 (a + l), where "a" is the first term and "l" is the last term.
- Sₙ can also be calculated using Sₙ = n/2 [2a + (n - 1)d], where "d" is the common difference.
Geometric Sequence
- A geometric sequence is a sequence where each term is multiplied by a constant to get the next term.
- Common Ratio in geometric sequence is calculated by dividing a term by its preceding term e.g. r = Tₙ₊₁ / Tₙ
- Example of a geometric sequence: sequence loga, loga², loga⁴,... is a geometric sequence with a common ratio of 2.
- In a geometric sequence, if r > 1 or a > 0, the sequence is increasing
- However, if r > 1 and a < 0, the sequence is decreasing
- For a sequence to be constant, r = 1
Terms of a Geometric Sequence
- The general term of a geometric sequence is described by Tₙ = arⁿ⁻¹, where "a" is the first term and "r" is the common ratio.
Geometric Means
- The geometric mean (g.m) of two numbers, a and b, is ±√(ab).
- The nth number inserted into a geometric mean is found with: (X₁, X₂, ...Xₙ)
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