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3 years ago

An arithmetic sequence has this recursive formula:

Mathematics

2 answers:

AnnZ [28]3 years ago

7 0

Answer:

Step-by-step explanation:

Sn=a(n-1) d

this formula for Calculating arithmetic sequence

prohojiy [21]3 years ago

4 0

Answer: D. $a_n=6+(n-1)(-3)$

Step-by-step explanation:

Given : An arithmetic sequence has this recursive formula:

$a_1=6 \ \ \; \ a_n=a_{n-1}-3$

Using the given information, Second term of arithmetic sequence will be :-

$a_2=a_{1}-3=6-3=3$

That means common difference = $a_2-a_1=3-6=-3$

The explicit formula for arithmetic sequence is given by :-

$a_n=a+(n-1)d$ , where a is the first term and d is the common difference.

Put a= 6 and d= -3, we get

The explicit formula for this sequence :-

$a_n=6+(n-1)(-3)$

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belka [17]

In the figure below

1) Using the theorem of similar triangles (ΔBXY and ΔBAC),

$\frac{BX}{BA}=\frac{BY}{BC}=\frac{XY}{AC}$

Where

$\begin{gathered} BX=4 \\ BA=5 \\ BY=6 \\ BC\text{ = x} \end{gathered}$

Thus,

$\begin{gathered} \frac{4}{5}=\frac{6}{x} \\ \text{cross}-\text{multiply} \\ 4\times x=6\times5 \\ 4x=30 \\ \text{divide both sides by the coefficient of x, which is 4} \\ \text{thus,} \\ \frac{4x}{4}=\frac{30}{4} \\ x=7.5 \end{gathered}$

thus, BC = 7.5

2) BX = 9, BA = 15, BY = 15, YC = y

In the above diagram,

$\begin{gathered} BC=BY+YC \\ \Rightarrow BC=15\text{ + y} \end{gathered}$

Thus, from the theorem of similar triangles,

$\begin{gathered} \frac{BX}{BA}=\frac{BY}{BC}=\frac{XY}{AC} \\ \frac{9}{15}=\frac{15}{15+y} \end{gathered}$

solving for y, we have

$\begin{gathered} \frac{9}{15}=\frac{15}{15+y} \\ \text{cross}-\text{multiply} \\ 9(15+y)=15(15) \\ \text{open brackets} \\ 135+9y=225 \\ \text{collect like terms} \\ 9y\text{ = 225}-135 \\ 9y=90 \\ \text{divide both sides by the coefficient of y, which is 9} \\ \text{thus,} \\ \frac{9y}{9}=\frac{90}{9} \\ \Rightarrow y=10 \end{gathered}$

thus, YC = 10.