16 Questions
In a geometric series, if the 1st term is 2 and the common ratio is 3, what is the 4th term?
27
For an arithmetic sequence, if the 5th term is 32 and the 9th term is 56, what is the common difference?
7
Find the sum of the first 7 terms of the arithmetic series where the 1st term is 3 and the common difference is 4.
101
Identify the type of sequence: 3, 6, 12, 24, ...
Geometric sequence
If the 11th term of an arithmetic sequence is 45 and the common difference is 4, what is the 21st term?
89
For a geometric series with a 1st term of 5 and a common ratio of 2, find the sum of the first 6 terms.
955
What is the defining characteristic of an arithmetic sequence?
Each term is created by adding or subtracting a definite number to the preceding number
In a geometric sequence, how is each term obtained?
By multiplying or dividing a definite number with the preceding number
What is the formula to find the nth term in an arithmetic sequence?
$a_n = a_1 + (n-1)d$
For a geometric sequence, what happens if the common ratio is greater than 1?
The terms increase at an increasing rate
What does the sum of all terms in a sequence form?
A series
What is the formula to compute the 21st term of an arithmetic sequence?
an = a1 + (n - 1)d
In an arithmetic sequence, if the common difference is 3 and the first term is 4, what is the 21st term?
61
Which term is referred to as the common ratio in a geometric series?
First term
What does the formula Sn = n(a1 + an)/2 represent in a series computation?
Sum of all terms
In a geometric series, what does it mean if the common ratio between terms is 0.5?
Each term is halved
Study Notes
Sequences and Series
- A sequence is an arrangement of objects or numbers in a particular order, following a specific rule.
- A sequence can be finite or infinite, depending on the number of terms.
- If 𝑎1, 𝑎2, 𝑎3, 𝑎4 … denotes the terms of the sequence, then 1, 2, 3, 4… denotes the position of the term.
Types of Sequences
- Arithmetic Sequence: every term is created by adding or subtracting a definite number to the preceding number.
- Geometric Sequence: every term is obtained by multiplying or dividing a definite number with the preceding number.
- Harmonic Sequence: if the reciprocals of all the elements of a sequence form an arithmetic sequence.
- Fibonacci Numbers: each element is obtained by adding two preceding elements, starting with 0 and 1.
Formulas
- Arithmetic Sequence: 𝑎𝑛 = 𝑎1 + (𝑛 − 1)𝑑
- Geometric Series: 𝑎𝑛 = 𝑎1 𝑟^(𝑛−1), 𝑆𝑛 = 𝑎1 (1 − 𝑟^n) / (1 − 𝑟)
- Harmonic Sequence: reciprocal of each term forms an arithmetic progression
Examples
- In an arithmetic sequence, the common difference is obtained by subtracting the first term from the second term.
- In a geometric series, the common ratio is obtained by dividing any two consecutive terms.
- In a harmonic sequence, the reciprocal of each term forms an arithmetic progression.
Test your knowledge of arithmetic and geometric series formulas with examples provided. Learn how to calculate the 21st term and sum of terms in a series using the given formulas.
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