Arithmetic Sequences & Series
Find the 10th term of arithmetic sequence: 3, 7, 11, 15, ...
a10 = 39
solution: d= 7 - 3 = 4
an = a1 + (n - 1)d
a10 = 3 + (10 - 1)4
a10 = 3 + 9*4
a10 = 3 + 36
a10 = 39
Common ratio of sequence: 3, 6, 18, 54,...?
3
solution: r = 6/2 = 3
What is the first term of the harmonic term if the first term of the arithmetic is 1/2
2
Solution: 1/1/2 = 2
The first five terms of the Fibonacci sequence, starting with 0 and 1
0, 1, 1, 2, 3
The 8th term of an arithmetic sequence is 50 and the common difference is 6. What is the first term?
a1 = 8
solution: an = a1 + (n - 1)d
given: a8 = 50 and d = 6
50 = a1 + (8 - 1)6
= a1 + 7*6
= a1 + 42
a1 = 50 - 42
a1 = 8
5th term of geometric sequence: 3, 6, 12, 24,...?
48
Solution: a5=3*24=3*16=48
The 3rd term of the harmonic sequence whose corresponding arithmetic sequence has terms: 1/3, 1/6, 1/9
9
The 9th term of the Fibonacci sequence starting with 0 and 1?
What is the sum of the first 20 terms of the sequences: 2, 5, 8, 11, ...?
answer: 610
solution: a1= 2, d = 3, n = 20
a20 = 2 + (20 - 1)3
= 59
Sn = 20/2 (2+59)
= 10 * 61
Sn = 610
What is the geometric mean between 5 and 45?
15
solution: sq/r(5⋅45)=sq/r(225)=15
(sq/r = square root)
The 1st term of a harmonic sequence is 3 and the 4th term is 1.5, what is the common difference of the corresponding arithmetic sequence?
1/9 or approximately 0.111...
Solution: d = 0.333.../3 = 0.111...
If the 4th term is 2 and the 5th term is 3, what is the 6th term?
5
Find the arithmetic mean between 12 and 20.
Answer: 16
solution: a+b/2 = 12+20/2 = 16.
The first term of a geometric sequence is 2, and the common ratio is 4. What is the 6th term?
2048
Solution: a6=2⋅45=2⋅1024=2048
DOUBLE JEOPARDY
Find the 3rd term of the harmonic sequence whose corresponding arithmetic sequence has first term 1/2 and common difference 1/4
1
b3=1/2+2×1/4=1/2+1/2=1
Which term of the Fibonacci sequence is the first to be greater than 20?
9th term (21)
If the sum of the first n terms of an arithmetic sequence is Sn=3n^2 + 5n, find the first term and common difference.
answer a1 = 8, d = 6
solution:
an = Sn - Sn-1.
a1 = S1 = 3(1)^2 + 5(1) - 8
a2 = S2 - S1 = (3(4) + 10) - 8 = 14.
d = 14-8 = 6.
The sum of the first 5 terms of the geometric sequence: 1, 2, 4, 8, 16?
31
solution:
S5=1⋅(1−25)/1−2=1⋅(1−32)/−1=−31/−1=31
Find the sum of the first 3 terms of a harmonic sequence whose corresponding arithmetic sequence starts at 1 and increases by 0.5 each term
13/6 or approximately 2.17
Solution S3=1+2/3+0.5=1+0.666...+0.5=2.166...
Find the sum of the first 6 terms of the Fibonacci sequence starting with 0 and 1.
12
Solution: 0 + 1 + 1 + 2 + 3 + 5 = 12