Mathematics, 02.02.2020 23:01 lasardia

for each recursively defined sequence below write the first few terms. then use the terms to write an explicit equation.

a1 = 17

an+1 = an – 3

a1 = 20

an+1 = 1/2 · an

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Answer from: laurydamas2002

Explicit formula for

1) a_n=17-3(n-1)

2) $a_n=(\frac{1}{2} )^{n-1}\cdot 20$

Step-by-step explanation:

W have to find the first few terms of the given sequence and then find an explicit equation

Explicit formula = f(n) + d(n-1) , where,

f(n) is first term,

d is common difference,

n-1 is one term less than the term number.

Given : $a_1=17\\\\a_{n+1}=a_n-3$

$\text{We put n =1, we get,}\\\\a_2=a_1-3=17-3=11=17-3(2-1)\\\\\\\text{We put n =2, we get,}\\\\\\a_3=a_2-3=11-3=8=17-3-3=17-3\cdot 2=17-3(3-1)\\\\\text{We put n =3, we get,}\\\\\\a_4=a_3-3=8-3=5=17-3-3-3=17-3\cdot 3=17-3(4-1)\\$

Thus, we obtained an explicit formula,

a_n=17-3(n-1)

Given : $a_1=20\\\\a_{n+1}=\frac{1}{2} \cdot a_n$

Thus, we obtained an explicit formula,

a_n=17-3(n-1)

$\text{We put n =1, we get,}\\\\\\a_2=\frac{1}{2} \cdot a_1=\frac{1}{2} \cdot 20=10=(\frac{1}{2} )^{2-1}\cdot 20\\\\\\\text{We put n =3, we get,}\\\\\\a_4=\frac{1}{2} \cdot a_3=\frac{1}{2} \cdot 5=\frac{5}{2}=(\frac{1}{2} )^{4-1}\cdot 20\\\\\\$

Thus, we obtained an explicit formula,

$a_n=(\frac{1}{2} )^{n-1}\cdot 20$

Answer from: shanyeah

1. Terms are 17, 14, 11, 8, 5,...... and explicit equation is $a_{n}=17-3(n-1)$ .

2. Terms are 20, 10, 5, 2.5, 1.25,...... and explicit equation is $a_{n}=20\times (\frac{1}{2})^{n-1}$ .

Step-by-step explanation:

Ques 1: We are given the recursive formula for the sequence as,

$a_{n+1}=a_{n}-3$ , where $a_{1}=17$ .

So, substituting the values of 'n' from {1,2,3,.....}, we get,

$a_{2}=a_{1+1}=a_{1}-3=17-3=14$

$a_{3}=a_{2+1}=a_{2}-3=14-3=11$

$a_{4}=a_{3+1}=a_{3}-3=11-3=8$

$a_{5}=a_{4+1}=a_{4}-3=8-3=5$

Thus, the sequence is given by 17, 14, 11, 8, 5,......

As, the explicit equation of an arithmetic sequence is of the form, $a_{n}=a_{1}+d(n-1)$ , where $a_{1}$ is the first term and 'd' is the common difference.

As, the common difference, d = 14 - 17 = -3

Thus, we get,

The given sequence has the explicit equation, $a_{n}=17-3(n-1)$ .

Ques 2: We are given the recursive formula for the sequence as,

$a_{n+1}=\frac{a_{n}}{2}$ , where $a_{1}=20$ .

So, substituting the values of 'n' from {1,2,3,.....}, we get,

$a_{2}=a_{1+1}=\frac{a_{1}}{2}=\frac{20}{2}=10$

$a_{3}=a_{2+1}=\frac{a_{2}}{2}=\frac{10}{2}=5$

$a_{4}=a_{3+1}=\frac{a_{3}}{2}=\frac{5}{2}=2.5$

$a_{5}=a_{4+1}=\frac{a_{4}}{2}=\frac{2.5}{2}=1.25$

Thus, the sequence is given by 20, 10, 5, 2.5, 1.25,......

As, the explicit equation of a geometric sequence is of the form, $a_{n}=a_{1}\times r^(n-1)$ , where $a_{1}$ is the first term and 'r' is the common ratio.