Geometric Sequences and Exponential Functions
Geometric sequences and exponential functions form the backbone of mathematical modeling, offering powerful tools to describe growth, decay, and patterns in various phenomena. In the realm of mathematics, these concepts intertwine, both showcasing the progression of values in a systematic, predictable manner. A geometric sequence, characterized by a common ratio between successive terms, unveils the elegance of multiplicative growth. Conversely, exponential functions, with their constant base raised to varying powers, depict rapid, sustained changes. Together, they illuminate the intricate relationships between numbers and reveal the fundamental principles governing natural and engineered systems alike.
Questions
- .What is #x# if the sequence #1,5, 2x+3# ....is an arithmetic sequence?
- For the sequence an = 3n - 5, what is the value of a10?
- What is the recursive formula for geometric sequence #{2,10,50,250,..........}#?
- What is the product of 9 and a number #t#?
- The sum of the first #5# terms of a geometric sequence is #93# and the #10#th term is #3/2#. What is the common ratio and what is the fourth term?
- How do you write an explicit rule for the sequence R= 2, a1= 5?
- What is the constant term in the expansion of #(x^2-x-2)^15#?
- What are the next three terms of the series #{4,24,144,.........}#?
- How do you tell whether the sequence -8, 2, 12.... is arithmetic, geometric, or neither?
- The first term of a geometric sequence with common ratio #7# is #1#. What is the 6th term?
- What is the next fraction in this sequence? Simplify your answer. 2/3, 1/3, 1/6, 1/12
- Find the sum of 5-7+8-9+11-11+14-13 up to 20 terms?
- What is Six more than half of a number x ?
- Consider the sequence 1, 3, 3, 3, 5, 5, 5, 5, 5, 7, 7, 7, 7, 7, 7, 7, ... What is its 2016th term?
- Sam invests $6000 in treasury notes and bonds. The notes pay 8% annual interest and the bonds pay 10% annual interest. If the annual interest is $550, how much is invested in bonds?
- How do you use the geometric sequence formula to find the nth term?
- How to find the sum of the first 14 terms of the arithmetic sequence a_5 =6, a_7=10?
- What is the formula for the #n#th term of the sequence #1, 3, 6, 10, 15,...# ?
- What's the next term in this sequence? #1/98, 1/49, 2/49, 4/49#
- How to show that #S_oo- S_n= (-1/3)^n# ,given, #S_oo=9/4# and #S_n=9/4(1-(-1/3)^n)#?