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UNIT 10  :  SEQUENCES AND SERIES

LESSON 1: ARITHMETIC SEQUENCES HOMEWORK QUESTIONS

Quick Review

Arithmetic Sequences:

A sequence such as 2, 3, 8, 13,  is called an Arithmetic Sequence.  These sequences have the following properties.

·        Terms are denoted as t1 , t2 , t3 , referring  to term1, term 2, term 3

·        The difference between successive terms is constant.  ie  t2  t1 = t3  t2 = t4  t3  etc

·        This difference is called the common difference and denoted using the letter d.  Here d = 5.

·        The first term is denoted using the letter a.  Here a = -2.

·        Successive terms are found by adding the common difference, d, to the preceding term.  Hence t5 = 13 + 5 = 18 etc.

·        The formula for the general term or nth term is   tn = a + (n  1)d

·        Arithmetic sequences are linear functions with domain the natural numbers N = {1, 2, 3, 4, 5, }

Homework Questions:

1. State which of the following are arithmetic.  Find a and d for those that are arithmetic.

2.  In each of the following arithmetic sequences, determine t10 and tn .

a)  -9, -2, 5, 12, 19,

b)  2, 2.5, 3, 3.5,

c)  x  2b, x, x + 2b, x + 4b,

3.  Find a, d, tn for the arithmetic sequences given the following terms.

4.  The third term of an arithmetic sequence is 14 and the ninth term is 1.  Find the twentieth term.

5.  Find the number of terms for the sequence below.

2, 9, 16, , 107

6.  Given the arithmetic sequence  5, 2, -1,  .  Which term is  82?

7.  Find the number of multiples of  7 between and including 21 and 147.

8.  Find the first term and common difference for the sequence defined by  tn = 5  2n.

9.  a)  Complete the table of values below for the sequence defined by tn = 2n  3.

 n 1 2 3 4 5 6 tn

b)  Use finite differences to explain why the sequence is arithmetic.

c)  Sketch the graph of n vs tn . What shape does the graph of an arithmetic sequence have?  Should you join the points on the graph?

10.  On the first day of practice, the track team ran 3 laps of the field.  The coach plans to increase the number of laps by two each practice.

How many laps will they run on the eleventh practice?

13.  Find the value of a that makes the following sequence arithmetic.

2a  1, 5, 3a + 1,

14.  Given an arithmetic sequence with tn = a + (n-1)d, prove that tn  tn-1 = d.

Solutions:

Solutions:

a)  t4  t3 = 5  3 = 2

t3  t2 = 3  1 = 2

t2  t1 = 1  (-1) = 2

The sequence has a common difference.  Hence it is arithmetic with a = -1 and d = 2.

b)  t4  t3 = -7  5 = - 12

t3  t2 = 5  (-3) = 8

t2  t1 = - 3  1 = - 4

The sequence does not have a common difference.  Hence it is not arithmetic.

The sequence does not have a common difference.  Hence it is not arithmetic.

d)  t4  t3 = 1  7 = -6

t3  t2 = 7  13 = -6

t2  t1 = 13  19 = -6

The sequence has a common difference.  Hence it is arithmetic with a = 19 and d = - 6.

e)  p, p  q, p  2q, p  3q,

t4  t3 = p  3q  (p  2q) = p  3q  p + 2q = - q

t3  t2 = p  2q  (p  q) = p  2q  p + q = - q

t2  t1 = p  q  p = - q

The sequence has a common difference.  Hence it is arithmetic with a = p  and d = - q.

2.  In each of the following arithmetic sequences, determine t10 and tn .

a)  -9, -2, 5, 12, 19,

b)  2, 2.5, 3, 3.5,

c)  x  2b, x, x + 2b, x + 4b,

Solutions:

a)      -9, -2, 5, 12, 19,

 a = -9 d = 7 t10 = ? tn = ?

b)  2, 2.5, 3, 3.5,

 a = 2 d = 0.5 t10 = ? tn = ?

c)  x  2b, x, x + 2b, x + 4b,

 a = x d = 2b t10 = ? tn = ?

3.  Find a, d, tn for the arithmetic sequences given the following terms.

4.  The third term of an arithmetic sequence is 14 and the ninth term is 1.  Find the twentieth term.

Solution:

5.  Find the number of terms for the sequence below.

2, 9, 16, , 107

Solution:

 a = 2 d = 7 n = ? tn = 107

6.  Given the arithmetic sequence  5, 2, -1,  .  Which term is  82?

Solution:

 a = 5 d = -3 n = ? tn = - 82

7.  Find the number of multiples of  7 between and including 21 and 147.

Solution:

 a = 21 d = 7 n = ? tn = 147

8.  Find the first term and common difference for the sequence defined by  tn = 5  2n.

Solution:

9.  a)  Complete the table of values below for the sequence defined by tn = 2n  3.

 n 1 2 3 4 5 6 tn

b)  Use finite differences to explain why the sequence is arithmetic.

c)  Sketch the graph of n vs tn . What shape does the graph of an arithmetic sequence have?  Should you join the points on the graph?

Solution:

a)

 n 1 2 3 4 5 6 tn -1 1 3 5 7 9

b)  The finite differences are:

Since the first finite differences are constant, the sequence is a linear function and hence arithmetic with common difference d = 2.

c)                     tn

n

The graph is linear, but only a series of isolated points as the domain of any sequence is the set of Natural Numbers {1, 2, 3, 4, }, not the Real Numbers.

-  ie  we cant have decimal or fractional values for n.

10.  On the first day of practice, the track team ran 3 laps of the field.  The coach plans to increase the number of laps by two each practice.  How many laps will they run on the eleventh practice?

Solution:

This forms an arithmetic sequence with first term 3 and common difference 2.  Find t11.

11.  The first 3 terms of an arithmetic sequence are 2, 3 and 8.  Is 561 a term of the sequence?  [Hint  -  Find tn]

Solution:

The first term is  2 and the common difference is 5.  Find tn.

Note:  If you try 563, n will come out to be 514 which is a natural number and hence 563 is a term of the sequence.

 a = 48 d = - 4 n = 9

13.  Find the value of a that makes the following sequence arithmetic.

2a  1, 5, 3a + 1,

Solution:

Recall to be arithmetic, the test is

14.  Given an arithmetic sequence with tn = a + (n-1)d, prove that tn  tn-1 = d.

Proof: