Master Arithmetic Sequences with Explicit and Recursive Rules

$n$	$a_{n}$	Rewrite
$1$	$a_{1}$	$a_{1} + 0 \cdot d$
$2$	$a_{2}$	$a_{1} + 1 \cdot d$
$3$	$a_{3}$	$a_{1} + 2 \cdot d$
$4$	$a_{4}$	$a_{1} + 3 \cdot d$
$5$	$a_{5}$	$a_{1} + 4 \cdot d$

n

a_{n}

Rewrite

1

a_{1}

a_{1} + 0 \cdot d

2

a_{2}

a_{1} + 1 \cdot d

3

a_{3}

a_{1} + 2 \cdot d

4

a_{4}

a_{1} + 3 \cdot d

5

a_{5}

a_{1} + 4 \cdot d

$a_{1} = 50, a_{n} = a_{n - 1} + 5$
Position	Substitute	Simplify
$n = 1$	$a_{1} = 50$	$a_{1} = 50$
$n = 2$	$a_{2} = a_{1} + 5 ⇓ a_{2} = 50 + 5$	$a_{2} = 55$
$n = 3$	$a_{3} = a_{2} + 5 ⇓ a_{3} = 55 + 5$	$a_{3} = 60$
$n = 4$	$a_{4} = a_{3} + 5 ⇓ a_{4} = 60 + 5$	$a_{4} = 65$

a_{1} = 50, a_{n} = a_{n - 1} + 5

Position

Substitute

Simplify

n = 1

a_{1} = 50

a_{1} = 50

n = 2

a_{2} = a_{1} + 5 ⇓ a_{2} = 50 + 5

a_{2} = 55

n = 3

a_{3} = a_{2} + 5 ⇓ a_{3} = 55 + 5

a_{3} = 60

n = 4

a_{4} = a_{3} + 5 ⇓ a_{4} = 60 + 5

a_{4} = 65

Determine whether the following sequence is arithmetic.

32, 30, 26, 20

When a sequence is arithmetic, the difference between two consecutive terms is constant. Examining our sequence, we can see that this is not the case. 32 -2 → 30 -4 → 26 -6 → 20 Therefore, the sequence is not arithmetic.

What are the next two terms of the following sequence?

2, 0, - 2, - 4

To describe the pattern of the sequence, we need to find the change that occurs between each pair of consecutive terms. Here, notice that the difference between each pair of consecutive terms in the sequence is - 2. 2-2 →0-2 →-2-2 →-4 To find the next two terms, we have to subtract 2 twice. 2-2 →0-2 →-2-2 →-4-2 → -6-2 → -8 The 5^\text{th} term is -6 and the 6^\text{th} term is -8.

Which formula represents the recursive rule for the following arithmetic sequence?

5.6, 6.1, 6.6, 7.1

A . B . C . D . A (n) = A (n - 1) + 0.5, A (1) = 5.6 A (n) = A (n - 1) - 0.5, A (1) = 5.6 A (n) = 2 A (n - 1) - 5.1, A (1) = 7.1 A (n + 1) = 2 A (n) - 5.1, A (1) = 7.1

In order to write the formula for the given arithmetic sequence, we first need to find the common difference. We can do that by subtracting the first term of the sequence from the second term. a_2-a_1=d ⇒ 6.1-5.6=0.5 The common difference is 0.5. To write the recursive rule, we will use the general form for a recursive rule of an arithmetic sequence. A(n)=A(n-1)+d In the formula, n is the term number and d is the common difference. By substituting the value for the common difference d into this formula, we can create our recursive rule.

The recursive rule for the arithmetic sequence is A(n)=A(n-1)+0.5, where A(1)=5.6. Therefore, the answer is option A.

Choose the correct explicit formula for the following recursive formula.

A (n) = A (n - 1) + 8, A (1) = - 1

A . B . C . D . A (n) = 8 - (n - 1) A (n) = 1 - (n - 1) 8 A (n) = 1 + (n - 1) 8 A (n) = - 1 + (n - 1) 8

The explicit formula of an arithmetic sequence combines the information provided by the two equations of the recursive form into a single equation. Recursive:& A(n)=A(n-1)+ d, & A(1)= a_1 [0.8em] Explicit:& A(n)= a_1+(n-1) d In these formulas, d is the common difference and a_1 is the first term. Looking at the given recursive formula, we can identify the common difference d and the value of the first term a_1. A(n)=A(n-1)+ 8, A(1)= -1 We can see that 8 is the common difference and the first term is -1. Now we have enough information to form an explicit formula for this sequence.

Therefore, the answer is option D.

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