Sequences and Series
1.
Find the common difference in an arithmetic sequence in which a10 – a20 = 70
a10 = a1 + 9d, a20 = a1 + 19d
70 = 9d – 19d = -10d, d = 7
2.
Find the sum: 1/e + 3/e + 5/e +…+21/e
=[1 + 3 + 5…+ 21]/e =11 [1 + 21]/(2e) = (11)(22)/(2e) = 121/e
3.
The sum of the first 12 terms in an arithmetic sequence is 156. What is the sum of the
1st and 12th terms?
12[a + a+11(d)]/2 = 156
6[2a + 11d]=156
[2a + 11d] = 26
4.
The fifth and 50th terms in an arithmetic sequence are 3 and 30. Find the first 4 terms.
a + 4d = 3
a + 49d = 30
45d = 27, d = 27/45 = 3/5.
What is a? a = 3-4(3/5) = (15-12)/5 = 3/5
Check: 3/5 + 49(3)/5 = 30 OK
first 3 terms are: 3/5, 6/5, 9/5 (5th is 3/5 + 12/5 = 15/5 = 3)
5.
Find the 29th term of the geometric sequence -1, 1, -1, 1, …
r= -1, a = -1
sum = =
𝑎(1−𝑟 𝑛 )
1−𝑟
= -1(1-(-1)29)/2 = (-1)(2/2) = -1
6.
Find the sum of the first 6 terms in the sequence -4, -2, -1…
geometric series, r = 1/2
sum = -4(1-(1/2)6)/(1/2)
(½)6 = 1/64; -4(63/64)(2) = -8(63/64) = -63/8
7.
Find the sum: ∑3𝑖=1 (2)
1 𝑖
r=1/2, a=1/2, sum = ½ (1-(1/2)3)/(1/2)= ½(7/8)(2) = 7/8
8.
Find the sum of the infinite geometric series: 9/10+ 9/100 + 9/1000 + …
r=1/10, a = 9/10
sum = (9/10)/(1-1/10) = 1
9.
Express the repeating decimal as a fraction in lowest terms: 0.432432432…
a = 0.432, r = 1/1000
sum = (0.432)/(999/1000) = 432/999 = 144/333 = 48/111 = 16/37