REVIEW OF ARITHMETIC SEQUENCES
The formula for the n th term a_{n} of an arithmetic sequence with a common difference d and a first term a_{1} is given by
a_{n} = a_{1} + (n  1 )d
The sum s_{n} of the first n terms of an arithmetic sequence is defined by
s_{n} = a_{1} + a_{2} + a_{3} + ... + a_{ n}
and is a_{1} is given by
s_{n} = n (a_{1} + a_{n}) / 2
Problem 1:
The first term of an arithmetic sequence is equal to 6 and the common difference is equal to 3. Find a formula for the n th term and the value of the 50 th term
Solution to Problem 1:

Use the value of the common difference d = 3 and the first term a_{1} = 6 in the formula for the n th term given above
a_{n} = a_{1} + (n  1 )d
= 6 + 3 (n  1)
= 3 n + 3

The 50 th term is found by setting n = 50 in the above formula.
a_{50} = 3 (50) + 3 = 153
Problem 2:
The first term of an arithmetic sequence is equal to 200 and the common difference is equal to 10. Find the value of the 20 th term
Solution to Problem 2:

Use the value of the common difference d = 10 and the first term a_{1} = 200 in the formula for the n th term given above and then apply it to the 20 th term
a_{20} = 200 + (10) (20  1 ) = 10
Problem 3:
An arithmetic sequence has a common difference equal to 10 and its 6 th term is equal to 52. Find its 15 th term.
Solution to Problem 3:

We use the n th term formula for the 6 th term, which is known, to write
a_{6} = 52 = a_{1} + 10 (6  1 )

The above equation allows us to calculate a_{1}.
a_{1} = 2

Now that we know the first term and the common difference, we use the n th term formula to find the 15 th term as follows.
a_{15} = 2 + 10 (15  1) = 142
Problem 4:
An arithmetic sequence has a its 5 th term equal to 22 and its 15 th term equal to 62. Find its 100 th term.
Solution to Problem 4:

We use the n th term formula for the 5 th and 15 th terms to write
a_{5} = a_{1} + (5  1 ) d = 22
a_{15} = a_{1} + (15  1 ) d = 62

We obtain a system of 2 linear equations where the unknown are a_{1} and d. Subtract the right and left term of the two equations to obtain
62  22 = 14 d  4 d

Solve for d.
d = 4

Now use the value of d in one of the equations to find a_{1}.
a_{1} + (5  1 ) 4 = 22

Solve for a_{1} to obtain.
a_{1} = 6

Now that we have calculated a_{1} and d we use them in the n th term formula to find the 100 th formula.
a_{100} = 6 + 4 (100  1 )= 402
Problem 5:
Find the sum of all the integers from 1 to 1000.
Solution to Problem 5:

The sequence of integers starting from 1 to 1000 is given by
1 , 2 , 3 , 4 , ... , 1000

The above sequence has 1000 terms. The first term is 1 and the last term is 1000 and the common difference is equal to 1. We have the formula that gives the sum of the first n terms of an arithmetic sequence knowing the first and last term of the sequence and the number of terms (see formula above).
s_{1000} = 1000 (1 + 1000) / 2 = 500500
Problem 6:
Find the sum of the first 50 even positive integers.
Solution to Problem 6:

The sequence of the first 50 even positive integers is given by
2 , 4 , 6 , ...

The above sequence has a first term equal to 2 and a common difference d = 2. We use the n th term formula to find the 50 th term
a_{50} = 2 + 2 (50  1) = 100

We now the first term and last term and the number of terms in the sequence, we now find the sum of the first 50 terms
s_{50} = 50 (2 + 100) / 2 = 2550
Problem 7:
Find the sum of all positive integers, from 5 to 1555 inclusive, that are divisible by 5.
Solution to Problem 7:

The first few terms of a sequence of positive integers divisible by 5 is given by
5 , 10 , 15 , ...

The above sequence has a first term equal to 5 and a common difference d = 5. We need to know the rank of the term 1555. We use the formula for the n th term as follows
1555 = a_{1} + (n  1 )d

Substitute a_{1} and d by their values
1555 = 5 + 5(n  1 )

Solve for n to obtain
n = 311

We now know that 1555 is the 311 th term, we can use the formula for the sum as follows
s_{311} = 311 (5 + 1555) / 2 = 242580
Problem 8:
Find the sum S defined by
10 
S = ∑ (2n + 1 / 2) 
n=1 
Solution to Problem 8:

Let us first decompose this sum as follows
10 
S = ∑ (2n + 1 / 2) 
n=1 
10 
10 
= 2 ∑ n 
+ ∑ (1 / 2) 
i=1 
n=1 

The term ∑ n is the sum of the first 10 positive integers. The 10 first positive integers make an arirhmetic sequence with first term equal to 1, it has n = 10 terms and its 10 th term is equal to 10. This sum is obtained using the formula s_{n} = n (a_{1} + a_{n}) / 2 as follows
10(1+10)/2 = 55

The term ∑ (1 / 2) is the addition of a constant term 10 times and is given by
10(1/2) = 5

The sum S is given by
S = 2(55) + 5 = 115
Exercises:
Answer the following questions related to arithmetic sequences:
a) Find a_{20} given that a_{3} = 9 and a_{8} = 24
b) Find a_{30} given that the first few terms of an arithmetic sequence are given by 6,12,18,...
c) Find d given that a_{1} = 10 and a_{20} = 466
d) Find s_{30} given that a_{10} = 28 and a_{20} = 58
e) Find the sum S defined by
20 
S = ∑ (3n  1 / 2) 
n=1 
f) Find the sum S defined by
20 
40 
S = ∑ 0.2 n 
+ ∑ 0.4 j 
n=1 
j=21 
Solutions to Above Exercises:
a) a_{20} = 60
b) a_{30} = 180
c) d = 24
d) s_{30} = 1335
e) 1380
f) 286
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