Arithmetic and geometry sequence

Sequence is an ordered list of numbers. The sum of the terms of a sequence is known as a series. While some sequences are merely random values, other sequences have a definite pattern that is used to obtain the sequence's terms. Two such sequences are the arithmetic and geometric sequences. Let's investigate the arithmetic sequence.

An arithmetic series is the sum of an arithmetic sequence. A geometric series is the sum of a geometric sequence. There are other forms of series, but you're unlikely to work with them till when you are doing n calculus. For reasons that will be explained in calculus, you can merely take the partial sum of an arithmetic sequence. The "partial" sum is the sum of a limited (that is finite) number of terms, like the first ten terms, or the fifth through the hundredth terms.

There is a formula for the first n terms of an arithmetic sequence, beginning with n = 1.

Find the common difference and the next term of the following sequence:

3, 11, 19, 27, 35,...

To obtain the common difference, you need to subtract a pair of terms. It doesn't matter which pair you select in so far as they're immediately next to each other:

11 – 3 = 8

19 – 11 = 8

27 – 19 = 8

35 – 27 = 8

The difference is at all time 8, therefore, d = 8. Then the next term is 35 + 8 = 43.

Obtain the common ratio and the seventh term of the following sequence:

2/9, 2/3, 2, 6, 18,...

To obtain the common ratio, I have to divide a pair of terms. It doesn't matter which pair you select in so far as they're just next to each other:

2/3⁄2//9=(2⁄3)(9⁄2)=9⁄3=3

2⁄2//3=(2⁄1)(3⁄2)=3⁄1=3

6⁄2=3

18⁄6=3

The ratio is constantly 3, therefore, r = 3. Then the sixth term is (18)(3) = 54 and the seventh term is (54)(3) = 162.

Due to the fact that arithmetic and geometric sequences are so good and regular, they have formulas.

For arithmetic sequences, the common difference is d, and the first term a1 is frequently known simply as "a". Due to the fact that you obtain the next term through the addition of the common difference, the value of a2 is merely a + d. The third term is a3 = (a + d) + d = a + 2d. The fourth term is a4 = (a + 2d) + d = a + 3d. Considering this pattern, the n-th term an will be made up of the form an = a + (n – 1)d.

For geometric sequences, the common ratio is r, and the first term a1 is often referred to simply as "a". Due to the fact that you obtain the next term by multiplying by the common ratio, the value of a2 is merely ar. The third term is a3 = r(ar) = ar2. The fourth term is a4 = r(ar2) = ar3. Following this pattern, the n-th term an will possess the form an = ar(n – 1).

Find the tenth term and the n-th term of the following sequence:

1/2, 1, 2, 4, 8,...

The differences are not equivalent: 2 – 1 = 1, but 4 – 2 = 2. Therefore, this isn't an arithmetic sequence. On the contrary, the ratios are the same: 2 ÷ 1 = 2, 4 ÷ 2 = 2, 8 ÷ 4 = 2. Therefore, this is a geometric sequence with common ratio r = 2 and a = 1/2. To obtain the tenth and n-th terms, we can merely plug into the formula an = ar(n – 1):

an = (1/2) 2n–1

a10 = (1/2) 210–1 = (1/2) 29 = (1/2)(512) = 256

Find the n-th term and the first three terms of the arithmetic sequence having a6 = 5 and d = 3/2.

The n-th term of an arithmetic sequence is of the form an = a + (n – 1)d. In this example, that formula offers us a6 = a + (6 – 1)(3/2) = 5. Solving this formula for the value of the first term of the sequence, you would obtain a = –5/2. Then:

a1 = –5/2, a2 = –5/2 + 3/2 = –1, a3 = –1 + 3/2 = 1/2, and an = –5/2 + (n – 1)(3/2)

Find the n-th term and the first three terms of the arithmetic sequence having a4 = 93 and a8 = 65.

Since a4 and a8 are four places apart, therefore, you would know from the definition of an arithmetic sequence that a8 = a4 + 4d. Making use of this, you can then solve for the common difference d:

65 = 93 + 4d 

–28 = 4d 

–7 = d

Again, we know that a4 = a + (4 – 1)d, therefore, making use of the value you just found for d, you can obtain the value of the first term a:

93 = a + 3(–7) 

93 + 21 = a 

114 = a

Immediately you have the value of the first term and the value of the common difference, you can plug-n-chug to obtain the values of the first three terms and the general form of the n-th term:

a1 = 114, a2 = 114 – 7 = 107, a3 = 107 – 7 = 100

an = 114 + (n – 1)(–7)

Find the n-th and the 26th terms of the geometric sequence with a5 = 5/4 and a12 = 160.

These two terms are 12 – 5 = 7 places apart, therefore, from the definition of a geometric sequence, we know that a12 = ( a5 )( r7 ). We can make use of this to solve for the value of the common ratio r:

160 = (5/4)(r7) 

128 = r7 

2 = r

Since a5 = ar4,

then we can solve for the value of the first term a:

5/4 = a(24) = 16a

Once you’ve obtained the value of the first term and the value of the common ratio, you can plug each into the formulas, and obtain your answers:

an = (5/64)2(n – 1)

a26 = (5/64)(225) = 2 621 440

The sum is, actually, n times the "average" of the first and last terms. This sum of the first n terms is called "the n-th partial sum".

Find the 35th partial sum of an = (1/2)n + 1

The 35th partial sum of this sequence is the sum of the first thirty-five terms. The first few terms of the sequence are:

a1 = (1/2)(1) + 1 = 3/2

a2 = (1/2)(2) + 1 = 2

a3 = (1/2)(3) + 1 = 5/2

The terms have a common difference of d = 1/2, therefore, this is indeed an arithmetic sequence. The last term in the partial sum will be a35 = a1 + (35 – 1) (d) = 3/2 + (34)(1/2) = 37/2. Then, puttng into the formula, the 35th partial sum is:

(n/2)(a1 + an) = (35/2)(3/2 + 37/2) = (35/2)(40/2) = 350

Deduce the value of n for which the following equation is true:

I know that the first term is a1 = 0.25(1) + 2 = 2.25. I can observe from the formula that everyone of the terms will be 0.25 units bigger than the term before them, therefore this is an arithmetical series. Then the summation formula for arithmetical series offers me:

(n/2)(2.25 + [0.25n + 2]) = 21 

n(2.25 + 0.25n + 2) = 42 

n(0.25n + 4.25) = 42 

0.25n2 + 4.25n – 42 = 0 

n2 + 17n – 168 = 0 

(n + 24)(n – 7) = 0

Solving the quadratic, we obtain that n = –24 (which won't work in this context) or n = 7.

n = 7

You could do the above exercise by adding terms till such a time you get to you get to the needed total of "21". But you may be asked to do a summation that needs for instance, eighty-six terms before you get the right total. Therefore, ensure that you can do the computations from the formula.

Obtain the sum of 1 + 5 + 9 + ... + 49 + 53.

Checking the terms, I can observe that this is indeed an arithmetic series: 5 – 1 = 4, 9 – 5 = 4, 53 – 49 = 4. I've got the first and last terms, but how many terms are there in total?

I have the n-th term formula, "an = a1 + (n – 1)d", and a1 has been given as = 1 and d = 4. Putting these into the formula, I can decipher how many terms there are:

an = a1 + (n – 1)d 

53 = 1 + (n – 1)(4) 

53 = 1 + 4n – 4 

53 = 4n – 3 

56 = 4n 

14 = n

Therefore, there are 14 terms in this series. Now we have obtained all the information we require:

1 + 5 + 9 + ... + 49 + 53 = (14/2)(1 + 53) = (7)(54) = 378

You can take the sum of a finite number of terms of a geometric sequence. And, for reasons you'll study in calculus, you can take the sum of an infinite geometric sequence, but only in the special circumstance that the common ratio r is between –1 and 1; that is, you have to have | r | < 1.

If a sequence of values follows a pattern of multiplying a fixed amount (not zero) times each term to arrive at the term after it, it is known as a geometric sequence. The number multiplied each time is constant (always the same). The fixed amount multiplied is known as the common ratio, r, meaning that the ratio (fraction) of the second term to the first term produces this common multiple. To obtain the common ratio, divide the second term by the first term.

Observe the non-linear nature of the scatter plot of the terms of a geometric sequence. The domain is made up of the counting numbers 1, 2, 3, 4, ... and the range is made up of the terms of the sequence. Whereas the x value increases by a constant value of one, the y value increases by multiples of two .

Geometric Sequence Common Ratio, r

5, 10, 20, 40, ... r = 2 multiply each term by 2 to obtain the next term or...divide a2 by a1 to get the common ratio, 2.

-11, 22, -44, 88, ... r = -2 multiply each term by -2 to obtain the next term .or...divide a2 by a1 to find the common ratio, -2.

4,8⁄3,16⁄9,32⁄27,64⁄81 r=2⁄3 multiply each term by 2/3 to obtain the next term or...divide a2 by a1 to find the common ratio, 2/3.

Formulas for geometric sequences and geometric series:

To obtain any term of a geometric sequence:

an=a·rn-1 where a1 is the first term of the sequence, r is the common ratio, n is the number of the term to find.

To find the sum of a certain number of terms of a geometric sequence: sn=a(1-rn)⁄1-r where Sn is the sum of n terms (nth partial sum), a1 is the first term, r is the common ration.

Note: a1 is often simply referred to as a.

Examples:

Problem Solution

1. Find the common ratio for the sequence

6,-3,3⁄2,-3⁄4,... 1. The common ratio, r, can be obtained by dividing the second term by the first term, which in this problem is -1/2. Cross-checking illustrates that multiplying each term by -1/2 yields the next entry.

2. Find the common ratio for the sequence given by the formula

an=5(3)n-1 2. The formula shows that 3 is the common ratio by its position in the formula. A listing of the terms will as well show what is happening in the sequence (start with n = 1). 5, 15, 45, 135, ... The list as well illustrates the common ratio to be 3.

3. Find the 7th term of the sequence 2, 6, 18, 54, ... 3. n = 7; a1 = 2, r = 3

ana1·rn-1

a72·37-1=1458 the seventh term is 1458

4. Find the 11th term of the sequence

1,-1⁄2,1⁄4,-1⁄8,... 4.n = 11; a1 = 1, r = -1/2

a11=1·(-1⁄2)11-1=1⁄1024

5. Solve the following: 5. The first few terms are –6, 12, –24, so this is a geometric series with common ratio r = –2.You can as well know that this must be a geometric series because of the form given for each term: since while the index increases, each term will be multiplied by an additional factor of –2.) The first term of the sequence is a = –6. Putting into the summation formula, you would obtain:

(-6)(1-(-2)20⁄1-(-2))

=(-6)(1-(-1048576)⁄1+2)

(-6)(-1048576⁄3)

=(-2)(-1048575)

=2097150