Write the General Term of each Arithmetic Sequence

The General Term Formula

Suppose the first term of an arithmetic sequence is u1 and the common difference is d. Then u2=u1+d, un=u1+2d, u4=u1+3d, and so on. Hence un=u1+(n-1)d. Term number n, the coefficient of d is one less than the term number (n-1).

For an arithmetic sequence with first term u1 and common difference d the general term or nth term is un=u1+(n-1)d.

Consider this arithmetic sequence: 3, 7, 11, 15, 19, 23, …
To determine an expression for the general term, tn, use the pattern in the terms. The common difference is 4. The first term is 3.

t1 3=3+4(0)
t2 3=3+4(1)
t3 3=3+4(2)
t4 3=3+4(3)
tn 3=3+4(n-1)

For each term, the second factor in the product is 1 less than the term number.
The second factor in the product is 1 less than n, or n-1.
write:

a general term of an arithmetic sequence a

The General Term of an Arithmetic Sequence
An arithmetic sequence with first term, t1, and common difference, d, is:

t1, t1+d, t1+2d, t1+3d, …

The general term of this sequence is: tn=t1+d(n-1)

(Worked Example) Example 1:
State which of the following are arithmetic sequences by finding the difference between successive terms. For those that are arithmetic, find the next term in the sequence, t4 and consequently find the functional definition for the nth term for the sequence, tn.

(a) t: {4, 9, 15, …} (b) t:{-2, 1, 4, …}

Solution:
(a) (1) To check that a sequence is arithmetic, see if a common difference exists.

9-4=5
15-9=6

(2) There is no common difference as 5≠6.

Since there is no common difference the sequence is not arithmetic.

(b) (1) To check that a sequence is arithmetic, see if a common difference exists.

1-(-2)=3
4-1=3

(2) The common difference is 3.

The sequence is arithmetic with the common difference d=3.

(3) The next term in the sequence, t4, can be found by adding 3 to the previous term, t3.

t4=t3+3
4+3=7

(4) To find the functional definition, write the formula for the nth term of the arithmetic sequence.

tn=a+(n-1)×d
=(ad)+nd

(5) Identify the values of a and d.

a=-2 and d=3

(6) Substitute a=-2 and d=3 into the formula and simplify.

tn=(-2-3)+n×3
tn=3n-5

Example 2. Find the 18th, 23rd and nth terms of the arithmetic progression.
-11, -9, -7, -5, …
Solution:
Here a1=-11, and d=-9+11=2
Thus,

a18=a1+(18-1)d
=-11+(17)(2)
=-11+34=23;
a23=a1+(23-1)d
=-11+(22)(2)
=-11+44=33; and
an=a1+(n-1)d
-11(n-1) (2)
=-11+2n-2=2n-13

Example 3: Write an Equation for the nth Term
Write an equation for the nth term of the arithmetic sequence 8, 17, 26, 35, … .

In this sequence, a1=8 and d=9. Use the nth term formula to write an equation.
Formula for nth term, an=a1+(n-1)d.

a1=8, d=9
an=8+(n-1)⋅9
an=8+9n-9
an=8+9n-9

An equation is an=8+9n-9.

Example 4. Find the formula for the nth term of the arithmetic sequence
1) 2, 5, 8, …
2) 107, 98, 89, … .
Solution:
1) Here a=2 and d=3, so

an=2+(n-1)×3=3n-1.

2) Here a=107 and d=-9, so
an=107+(n-1)×(-9)=116-9n.

We can also check whether a given number belongs to a given arithmetic sequence.

Example 5:

write an expression for the nth term of the sequence

a) 2, 4, 6, 8, 10, …
Answer: an=2n

b) 3, 5, 7, 9, 11, 13, …

I know I need a 2n since each term is 2 apart.
Solve 2(1)-x=3 to figure out the number that goes after the 2n.
Answer: an=2n+1

c) -3, -1, 1, 3, 5, …

Each term is 2 apart, so again I need a 2n.
Solve 2(1)-x=-3 to figure out what to write after the 2n.
Answer: an=2n – 5

d) 6, 10, 14, 18, 22, …

Each term is 4 apart, so I need a 4n.
Solve 4(1)+x=6 to figure out what number goes after the 4n.
Answer: an=4n+2

e) 7, 13, 19, 25, …

Each term is 6 apart, so I need a 6n.
Solve 6(1)+x=7 to figure out what number goes after the 6n.
Answer: an=6n+1

Example 6: Find a formula for the nth term of the arithmetic sequence with initial term a1 and common difference d.

a) a1=4; d=3

I know I need a 3n for the common difference to be 3. I just need to solve for the number after the 3n.

an=3n+x
a1=3(1)+x
4=3+x
1=x

Answer: an=3n+1

b) a1=-2; d=1

I need a 1n, for the common difference of 1. I need to solve for the number after the 1n.

an=1n+x
a1=1(1)+x
-2=1+x
-3=x

Answer: an=1n-3 orjust n-3
c) a1=4; d=-3

I need a -3n, for the common difference of -3. I need to solve for the number after the -3n.

an=-3n+x
a1=-3(1)+x
4=-3+x
7=x

Answer: an=-3n+7

d) a1=-2; d=-1

I need a -1n, for the common difference of -1. I need to solve for the number after the -1n.

an=-1n+x
a1=-1(1)+x
-2=-1+x
-1=x

Answer: an=-1n-1 or just –n-1

Example 7. Consider the sequence 2, 9, 16, 23, 30, …
a) Show that the sequence is arithmetic.
b) Find a formula for the general term un.
c) Find the 100th term of the sequence.
d) Is (i) 828 (ii) 2341 a term of the sequence?
solution:
a) 9-2=7

16-9=7
23-16=7
30-23=7

The difference between successive terms is constant. So, the sequence is arithmetic, with u1=2 and d=7.
b) un=u1+(n-1)d
un=2+7(n-1)
un=7n-5

c) If n=100, u100=7⋅100-5=700-5=695
d) (i) Let un=828
7n-5=828
7n=833
n=119

∴ 828 is a term of the sequence, and in fact is the 119th term.
(ii) Let un=2341
7n-5=2341
7n=2346
n=335+1/7

But it must be an integer, so 2341 is not a member of the sequence.

Example 8. Find the general term un for an arithmetic sequence with u3=8 and u8=-17.
Solution:
Using un=u1+(n-1)d.

u3=8∴u1+2d=8…(1)
u8=-17∴u1+7d=-17…(2)

Eliminate u1 to get d by (1)-(2).
eliminate a

(1)…u1+2(-5)=8→u1=8+10=18
Now un=u1+(n-1)d.
un=18-5(n-1)
un=18-5n+5
un=23-5n

Check:
u3=23-5⋅3=23-15=8✓
u8=23-5⋅8=23-40=-17✓

Let’s read post Write an equation or explicit formula for the nth term of each geometric sequence.

Example 9. Fill in the blanks in the following table, given that a is the first term, d the common difference and an the nth term of the AP:

a d n an
(i) 7 3 8
(ii) -18 10 0
(iii) -3 18 -5
(iv) -18.9 2.5 3.6
(v) 3.5 0 105

Solution:
(i) Here, a=7, d=3 and n=8, so, putting the values in an=a+(n-1)d, we get:

an=7+(8-1)(3)
an=7+21=28

(ii) Here, a=-18, n=10 and an=0, so, putting the values in an=a+(n-1)d, we get:

0=-18+(10-1)d
⇒18=9d ⇒d=2

(iii) Here, d=-3, n=18 and an=-5, so, putting the values in an=a+(n-1)d, we get:

5=a+(18-1)(-3)
⇒-5=a-51 ⇒a=46

(iv) Here, a=-18.9, d=2.5 and an=3.6, putting the values in an=a+(n-1)d, we get:

3.6=-18.9+(n-1)(2.5)
⇒3.6=-18.9+2.5n-2.5
⇒2.5n=3.6+21.4=25.0
n=10

(v) Here, a=3.5, d=0 and n=105, putting the values in an=a+(n-1)d, we get

an=3.5+(105-1)(0)

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