Geometric Sequence; Arithmetic Sequence w/o 2,3,7

In summary, a geometric sequence is a sequence of numbers where each term is found by multiplying the previous term by a constant ratio. An arithmetic sequence without 2, 3, or 7 is a sequence of numbers where each term is found by adding a constant value to the previous term, with the exclusion of the values 2, 3, and 7. The main difference between these two types of sequences is the method of finding each term. A sequence cannot be both geometric and arithmetic without the excluded values, as the definitions for each type of sequence require a constant ratio or value respectively. The formula for finding the nth term of a geometric sequence is <i>a<sub>n</sub> = a<sub>1</
  • #1
mustang
169
0
Problem 8.
Find x & y if the sequence 2y, 2xy, 2, xy/2,...is geometric.

Problem 9.
Find an arithmeitc sequence none of whose terms are divisible by 2, 3, or 7.

Prtoblem 10.
Consider two arithmetic sequences:
A:3, 14, 25.. B: 2, 9 , 16, ...
Write the first five terms of sequence A that are also terms of sequence B.

Problem 11b. Given two terms of a geometric sequence, under what conditions will there be two different common ratios that could be used to find two sequences that have the given terms?

Problem 16. Show that 1 + 2 + 4 +...+2^(n-1)=2^(n)-1
 
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  • #2
Yep, looks like Algebra 2 problems.
 
  • #3


Problem 8. To find x and y, we can use the formula for a geometric sequence: a_n = a_1 * r^(n-1). In this case, a_1 = 2y and a_2 = 2xy, so we have 2xy = (2y) * r. Similarly, a_3 = 2 and a_4 = xy/2, giving us xy/2 = 2 * r^2. Solving these equations simultaneously, we get x = 2 and y = 4.

Problem 9. An example of an arithmetic sequence that does not have terms divisible by 2, 3, or 7 is: 5, 11, 17, 23, 29, ...

Problem 10. The first five terms of sequence A that are also terms of sequence B are: 9, 16, 23, 30, 37.

Problem 11b. There will be two different common ratios if the two given terms are not consecutive terms in the geometric sequence. In other words, if the common difference between the two given terms is not a multiple of the common ratio.

Problem 16. We can use mathematical induction to prove this statement.
Base case: When n = 1, the left side is equal to 1, and the right side is equal to 2^(1) - 1 = 1. So the statement holds true for n = 1.
Inductive step: Assume the statement holds true for n = k, i.e. 1 + 2 + 4 + ... + 2^(k-1) = 2^(k) - 1.
For n = k+1, we have 1 + 2 + 4 + ... + 2^(k-1) + 2^(k) = 2^(k+1) - 1.
Adding 2^(k) on both sides, we get 1 + 2 + 4 + ... + 2^(k-1) + 2^(k) + 2^(k) = 2^(k+1) - 1 + 2^(k).
Simplifying, we get 1 + 2 + 4 + ... + 2^(k) + 2^(k) = 2^(k+1) -
 

What is a geometric sequence?

A geometric sequence is a sequence of numbers where each term is found by multiplying the previous term by a constant ratio. For example, the sequence 1, 2, 4, 8, 16 is a geometric sequence with a common ratio of 2.

What is an arithmetic sequence without 2, 3, or 7?

An arithmetic sequence without 2, 3, or 7 is a sequence of numbers where each term is found by adding a constant value to the previous term. However, in this case, the values 2, 3, and 7 are excluded from the sequence.

What is the difference between a geometric sequence and an arithmetic sequence without 2, 3, or 7?

The main difference between a geometric sequence and an arithmetic sequence without 2, 3, or 7 is the method of finding each term. In a geometric sequence, each term is found by multiplying the previous term by a constant ratio, while in an arithmetic sequence without 2, 3, or 7, each term is found by adding a constant value to the previous term.

Can a sequence be both geometric and arithmetic without 2, 3, or 7?

No, a sequence cannot be both geometric and arithmetic without 2, 3, or 7. This is because the definition of a geometric sequence involves a constant ratio, while the definition of an arithmetic sequence involves a constant value. Without the values 2, 3, and 7, it is not possible to have a common ratio or value to satisfy both definitions.

What is the formula for finding the nth term of a geometric sequence?

The formula for finding the nth term of a geometric sequence is an = a1 * rn-1, where an represents the nth term, a1 represents the first term, and r represents the common ratio.

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