Arithmetic Sequences & Series
Given the sequence (2, 6, 10, 14, ...), find the next three terms.
Solution: The common difference is (+-4). Next terms:(18, 22, 26).
Identify the common ratio (r) for the sequence: 2, -6, 18, -54, ....
Solution: r = -3
Determine if the following sequence is arithmetic, geometric, or neither: 1,4,9,16,25,....
Solution: Neither (It is the sequence of perfect squares)
State the explicit formula used to find the nth term (an) of an arithmetic sequence.
Solution: an=a1+(n-1)d
An arithmetic series had 15 terms. The first term is 2 and the last term is 44. Find the sum of the entire series.
Solution: 345 (Using S15 = 15/2(2+44))
Given the sequence (5, 11, 17, 23,....), find the next two terms.
The common difference is (d = 11 - 5 = 6). The next terms are (23 + 6 = 29) and (29 + 6 = 35). 29 and 35.
Find the 6th term of the geometric sequence: 5, 10, 20, 40, .....
Solution: 160 (Using a6=5(2)5)
Insert two arithmetic means between 5 and 20.
Solution: 10 and 15 (The sequence is 5,10,15,20)
State the formula for the sum of finite geometric series (Sn)
Solution: Sn = a1(1-r2)/1 - r or Sn = a1(rn-1 )/ r-1
Find the smallest value of n such that the sum of the geometric series 1 + 2 + 4 + 8 + .... exceeds 1,000.
Solution: n = 10 (The sum formula yields 2n-1>1000; 210 - 1 = 1023)
Find the (25th) term of the arithmetic sequence (7, 4, 1, -2,.....)
Use an= a1 + (n-1)d with a1 = 7, d = -3, and n = 25
a25 = 7 + (25 - 1)(-3)
= 7 + 24(-3)
= 7 - 72
= -65
Evaluate the sum of the infinite geometric series: 12 + 4 + 4/3 + 4/9 + ....
Insert two geometric means between 3 and 24.
Solution: 6 and 12 (The sequence is 3,6,12,24)
What condition must the common ratio r satisfy for an infinite geometric series to converge (have a finite sum)?
Solution: lrl<1(or -1<r<1)
Three numbers form an arithmetic sequence. Their sum is 15, and their product is 80. Find the three numbers.
Solution: 2,5,8(or 8,5,2)
How many terms are in the finite arithmetic sequence 12, 19, 26,....,180?
Solution: Use an = a1 + (n-1)d with an = 180, a1 = 12, and d = 7
180 = 12 + (n - 1)7
168 = 7(n - 1)
24 = n - 1
n = 25
A geometric series has a first term of 3 and a sum to infinity of 4. What is its common ratio r?
Solution: r = 1/4 (From 4 = 3/1-r)
Express the repeating decimal 0.444... as a fraction using the concept of infinite geometric series.
Solution: 4/9 (Where a1 = 4/10 and r = 1/10)
Write the formula for the sum of an infinite geometric series (Sinfinity), and state the condition required for the series to converge.
Solution: Sinfinity = a1/1-r
Find the sum of all multiples of 3 between 10 and 100.
Solution: 1,665
In an arithmetic sequence, the 5th term is 19 and the 11th term is 43. Find the first terms a1 and the common difference d.
Solution: Set up a system:
a5 = a1 + 4d = 19
a11 = a1 + 10d = 43
Substract the first equation from the second: 6d = 24, d = 4
a1 + 4(4) = 19
a1 = 3
Find the exact sum of the first 6 terms of the geometric series: 2 - 6 + 18 - 54 + ....
Solution: -364 (Using S6= 2(1-(-3)6)/1-(-3))
The numbers x, x+3 and x+9 form the first three terms of a geometric sequence. Find the value of x.
Solution: x = 3 (Set up the ratio: (x+3)/x = (x+9)/(x+3) and solve)
Write a recursive formula for the sequence: 3,6,12,24,...
Solution: a1 = 3, an = 2an-1 for n greater than equal to 2
A ball is dropped from a height of 10 meters. Each time it hits the ground, it rebounds to 3/4 of its previous height. Find the total vertical distance the ball travels before coming to a rest.
Solution: 70 meters (Account for the initial drop of 10, plus two times the infinite bounce series: 10+2(7.5/1-0.75) = 10+60