MATH 140 Class Exercise on Sequences Names__________________
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1. Arithmetic Sequences.
We found that in an arithmetic sequence with initial term a and common
difference d the nth term is always given by 
.
a) Find the 100th term of the arithmetic sequence that begins 5,12,19,… ___________
b) How many terms are there in the sequence 5,12,19, …,726 ? __________
c) We developed the following method to sum arithmetic
sequences. Suppose we wish to add the
following sequence: 
. We actually write
out the sum twice in opposite orders and add vertically:
Thus twice the sum, 2S, is 68 times the number of terms in the sequence. There are 11 terms. 2S = 11x68 = 748. Thus the original sum is 374.
Find 
____________
d) In this part we are concerned with the sequence 1,3,5,7,.., the arithmetic sequence of consecutive odd numbers starting at 1.
What is the nth term (in terms of n) of the sequence 1, 3, 5, …? ___________
Let 
be the nth term you
found. Now apply our technique to find
the sum of 
. Your answer should be
in terms of n.
2. Triangular numbers
The triangular
numbers, 
, were numbers of the form 
which could be formed into an equilateral triangle. For
example 
. 6 is the third triangular number.
Compute the following:
What pattern do you notice?__________________________________
Based on that pattern what is 
Compute:
What pattern do you find?__________________
Compute 
using this
pattern. ________________
3. Geometric Sequences
A geometric
sequence is a sequence of the form 
. For example 
is a geometric sequence with a = 2 and r = 3. a is called the initial term and r is called
the common quotient. We can rewrite the sequence as 
.
a) Write the first five terms of the geometric sequence with a = 5 and r = 2.
_____,_____,_____,_____,_____
b) The formula for the nth term of a geometric sequence is 
. Note the n-1 term
which is just like the term in the arithmetic sequences.
Let a = 7 and r = 3. Find the fifth term in the geometric sequence determined by a and r.
c) Geometric sequences have sums also. The formula for such a sum is actually easier to find than for arithmetic sequences. Write out the first 6 terms of a geometric sequence with a = 1 and r = 2.
We look at longer and longer sums from this sequence:
Look for a pattern in the sums based on the number of terms being added.
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d) The sum of the first n terms of the geometric sequence
given by a and r is 
. Check this formula
for the sums in part c).
e) Find the sum of the first 6 terms of the geometric series given by a = 1 and r = 3.
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