Mathematics: write a recursive rule for the nth term of the sequence: (3.2, 4.7, 6.2.)

write a recursive rule for the nth term of the, №9614019, 17.05.2023 22:45

Mathematics

Write the recursive rule for the nth term. Then find the 5th term. {9, 14, 19...]

Step-by-step answer

P Answered by PhD

1

Step-by-step explanation:

{9, 14, 19, ...} so it looks like 5 is added to the previous term to get next

The recursive rule may be

$a_{1} =9$

$a_{n} = a_{n-1} +5$

The fith term is you find by writing the arithmetic sequence formula

$a_{n} = a_{1} + (n-1) d$ , where d is how much you add

yet for the 5th term

$a_{5} = a_{1} + (5-1)* 5$

$a_{5} = 9 +4*5 = 9+20 =29$

Mathematics

Write the recursive rule for the nth term. Then find the 5th term. {9, 14, 19...]

Step-by-step answer

P Answered by PhD

1

Step-by-step explanation:

{9, 14, 19, ...} so it looks like 5 is added to the previous term to get next

The recursive rule may be

$a_{1} =9$

$a_{n} = a_{n-1} +5$

The fith term is you find by writing the arithmetic sequence formula

$a_{n} = a_{1} + (n-1) d$ , where d is how much you add

yet for the 5th term

$a_{5} = a_{1} + (5-1)* 5$

$a_{5} = 9 +4*5 = 9+20 =29$

Mathematics

Find a recursive rule for the nth term of the sequence. 5, 20, 80, 320, ...

Step-by-step answer

P Answered by Master

Hello,

$U_1=5\\U_{(n)}=4*U{(n-1)}$

Mathematics

A geometric sequence is given by the recursive rule $a_1= frac{2}{3}, a_n=3a_{n-1}$ . Give an explicit rule for the nth term of the sequence $b_1, b_2, b_3, dots ,$ given that $b_1=a_1, b_2=a_3, b_3=a_5, dots$ .

Step-by-step answer

P Answered by Specialist

$b_n = \frac{2}{27} (9^n)$

Step-by-step explanation:

Given

$a_1=\frac{2}{3},\ a_n=3a_{n-1}.$

Required

An explicit rule for $b_1,\ b_2,\ b_3,\ \dots$

Where $b_1=a_1,\ b_2=a_3,\ b_3=a_5,\ \dots$

We have:

$a_1=\frac{2}{3},\ a_n=3a_{n-1}.$

Calculate a2

$a_2 = 3a_{2-1}$

a_2 = 3a_1

$a_2 = 3 * \frac{2}{3}$

a_2 = 2

Calculate a3

$a_3 = 3a_{3-1}$

a_3 = 3a_2

a_3 = 3 *2

a_3 = 6

Calculate a4

$a_4 = 3a_{4-1}$

a_4 = 3a_3

a_4 = 3*6

a_4 = 18

Calculate a5

$a_5 = 3a_{5-1}$

a_5 = 3a_4

a_5 = 3*18

a_5 = 54

So:

$b_1=a_1,\ b_2=a_3,\ b_3=a_5,\ \dots$

$a_1=\frac{2}{3}$ a_3 = 6 a_5 = 54

The above sequence form a geometric sequence.

Calculate common ratio (r)

$r = \frac{b_3}{b_2}$

$r = \frac{a_5}{a_3}$

$r = \frac{54}{6}$

r = 9

So, the explicit formula is:

$b_n = b_1 * r^{n-1$

b_1=a_1 , so:

$b_n = a_1 * r^{n-1$

$a_1=\frac{2}{3}$ , so:

$b_n = \frac{2}{3} * 9^{n-1$

Split:

$b_n = \frac{2}{3} * \frac{9^n}{9}$

$b_n = \frac{2}{3*9} (9^n)$

$b_n = \frac{2}{27} (9^n)$

The explicit rule is: $b_n = \frac{2}{27} (9^n)$

Mathematics

A geometric sequence is given by the recursive rule $a_1= frac{2}{3}, a_n=3a_{n-1}$ . Give an explicit rule for the nth term of the sequence $b_1, b_2, b_3, dots ,$ given that $b_1=a_1, b_2=a_3, b_3=a_5, dots$ .

Step-by-step answer

P Answered by Master

$b_n = \frac{2}{27} (9^n)$

Step-by-step explanation:

Given

$a_1=\frac{2}{3},\ a_n=3a_{n-1}.$

Required

An explicit rule for $b_1,\ b_2,\ b_3,\ \dots$

Where $b_1=a_1,\ b_2=a_3,\ b_3=a_5,\ \dots$

We have:

$a_1=\frac{2}{3},\ a_n=3a_{n-1}.$

Calculate a2

$a_2 = 3a_{2-1}$

a_2 = 3a_1

$a_2 = 3 * \frac{2}{3}$

a_2 = 2

Calculate a3

$a_3 = 3a_{3-1}$

a_3 = 3a_2

a_3 = 3 *2

a_3 = 6

Calculate a4

$a_4 = 3a_{4-1}$

a_4 = 3a_3

a_4 = 3*6

a_4 = 18

Calculate a5

$a_5 = 3a_{5-1}$

a_5 = 3a_4

a_5 = 3*18

a_5 = 54

So:

$b_1=a_1,\ b_2=a_3,\ b_3=a_5,\ \dots$

$a_1=\frac{2}{3}$ a_3 = 6 a_5 = 54

The above sequence form a geometric sequence.

Calculate common ratio (r)

$r = \frac{b_3}{b_2}$

$r = \frac{a_5}{a_3}$

$r = \frac{54}{6}$

r = 9

So, the explicit formula is:

$b_n = b_1 * r^{n-1$

b_1=a_1 , so:

$b_n = a_1 * r^{n-1$

$a_1=\frac{2}{3}$ , so:

$b_n = \frac{2}{3} * 9^{n-1$

Split:

$b_n = \frac{2}{3} * \frac{9^n}{9}$

$b_n = \frac{2}{3*9} (9^n)$

$b_n = \frac{2}{27} (9^n)$

The explicit rule is: $b_n = \frac{2}{27} (9^n)$

Mathematics

Write a rule to find the nth term for an arithmetic sequence given the following: a3 = 14 a12 = 59 recursive rule: ? explicit rule: ? write an explicit and recursive rule for a sequence given the following: a4 = 2 r = 1/3 recursive rule: ? explicit rule: ?

Step-by-step answer

P Answered by PhD

ANSWER

See explanation

EXPLANATION

Question 1:

The third term of the arithmetic sequence is :

14=a+2d...(1)

The twelveth term is

59=a+11d...(2)

Subtract equation (1) from (2)

45=9d

This implies that

d=5

a=14-2(5)=4

The explicit rule is;

$a_{n}=4 + 5(n - 1)$

$a_{n}=4 + 5n -5$

$a_{n} = 5n -1$

Recursive formula:

$a_{n}=a_{n - 1} + 5$

Question 2

The geometric sequence has the fourth term to be 2 and the common ratio to be r=⅓

This implies that,

$a {( \frac{1}{3} })^{3} = 2$

This implies that,

$\frac{a}{27} = 2$

a = 54

The explicit rule:

$a_n=54 {( \frac{1}{3} })^{n - 1}$

The recursive rule is

$a_n=( \frac{1}{3} )a_{n-1}$

where,

a_1 = 54

Mathematics

Write a rule to find the nth term for an arithmetic sequence given the following: a3 = 14 a12 = 59 recursive rule: ? explicit rule: ? write an explicit and recursive rule for a sequence given the following: a4 = 2 r = 1/3 recursive rule: ? explicit rule: ?

Step-by-step answer

P Answered by PhD

ANSWER

See explanation

EXPLANATION

Question 1:

The third term of the arithmetic sequence is :

14=a+2d...(1)

The twelveth term is

59=a+11d...(2)

Subtract equation (1) from (2)

45=9d

This implies that

d=5

a=14-2(5)=4

The explicit rule is;

$a_{n}=4 + 5(n - 1)$

$a_{n}=4 + 5n -5$

$a_{n} = 5n -1$

Recursive formula:

$a_{n}=a_{n - 1} + 5$

Question 2

The geometric sequence has the fourth term to be 2 and the common ratio to be r=⅓

This implies that,

$a {( \frac{1}{3} })^{3} = 2$

This implies that,

$\frac{a}{27} = 2$

a = 54

The explicit rule:

$a_n=54 {( \frac{1}{3} })^{n - 1}$

The recursive rule is

$a_n=( \frac{1}{3} )a_{n-1}$

where,

a_1 = 54

Mathematics

Can someone match the terms to the right definition. 1 a rule for generating terms of a sequence that depends on one or more previous terms of the sequence. 2 a rule that can be used to find the nth term of a sequence without calculating previous terms of the sequence. also called the explicit rule. 3 a value that is held constant in a specific function, or model, while other variables might change. 4 the constant d of the sequence. 5 a sequence where the difference between consecutive terms is a constant. 6 a variable that takes on different values

Step-by-step answer

P Answered by Specialist

11

recursive rule—a rule for generating terms of a sequence that depends on one or more previous terms of the sequence

iterative rule—rule that can be used to find the nth term of a sequence without calculating previous terms of the sequence, also called the Explicit Rule

model parameter—value that is held constant in a specific function, or model, while other variables might change

common difference—the constant d of the sequence.

arithmetic sequence—a sequence where the difference between consecutive terms is a constant

model variable—a variable that takes on different values in a specific function, or model, while other values are held constant

common ratio—the constant r of the sequence

sequence—a list of numbers, finite or infinite, that follow a particular pattern

terms of a sequence—the numbers in a sequence

geometric sequence—a sequence where the ratio between consecutive terms is a constant

1. C 2. D 3. I 4. G 5. E 6. J 7. H 8. A 9. B 10. F

Mathematics

Can someone match the terms to the right definition. 1 a rule for generating terms of a sequence that depends on one or more previous terms of the sequence. 2 a rule that can be used to find the nth term of a sequence without calculating previous terms of the sequence. also called the explicit rule. 3 a value that is held constant in a specific function, or model, while other variables might change. 4 the constant d of the sequence. 5 a sequence where the difference between consecutive terms is a constant. 6 a variable that takes on different values

Step-by-step answer

P Answered by Master

11

recursive rule—a rule for generating terms of a sequence that depends on one or more previous terms of the sequence

iterative rule—rule that can be used to find the nth term of a sequence without calculating previous terms of the sequence, also called the Explicit Rule

model parameter—value that is held constant in a specific function, or model, while other variables might change

common difference—the constant d of the sequence.

arithmetic sequence—a sequence where the difference between consecutive terms is a constant

model variable—a variable that takes on different values in a specific function, or model, while other values are held constant

common ratio—the constant r of the sequence

sequence—a list of numbers, finite or infinite, that follow a particular pattern

terms of a sequence—the numbers in a sequence

geometric sequence—a sequence where the ratio between consecutive terms is a constant

1. C 2. D 3. I 4. G 5. E 6. J 7. H 8. A 9. B 10. F

Mathematics

a sequence has a first number of 3 and a half common difference of 4 what is the recursive rule of nth term

Step-by-step answer

P Answered by PhD

f(n) = f(n-1) + 8 for n 1