The first term of a geometric sequence is −2 and the common ratio is 3. What is the fourth term of the sequence?

Mathematics

1 answer:

Elden [556K]3 years ago

6 0

Answer:

Step-by-step explanation:

The formula for a geometric sequence is

$a_n=a_1*r^{n-1}$ where n is the position in the sequence, a1 is the first number in the sequence, and r is the common ratio. For us, a1 = -2 and r = 3, n = the 4th term:

$a_4=-2*3^{4-1}$ and

$a_4=-2*3^3$ and

$a_4=-2*27$ so

$a_4=-54$

The 4th term in the geometric sequence is -54.

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Answer: 117.6° ; 32.4° .
_________________________
Explanation:
_______________________________________________________
Note: ALL triangles, by definition, have exactly 3 (THREE) sides and exactly 3 (THREE) angles.
_____________________
We are given the following:
_______________________
We have a triangle.

Angle 1: m∡1 = (8x) ;

Angle 2: m∡2 = (2x + 3) ;

Angle 3: m∡3 = 30.
______________________________________
We are asked to find: "m∡1" and " m∡2" .
______________________________________
Note: In ALL TRIANGLES, the measurements of all THREE (3) angles ALWAYS add up to 180 degrees.
________________________________________
So, " m∡1 + m∡2 + m∡3 = 180 " .
________________________________________
Let us substitute our given values for the measurements in EACH of
the THREE (3) angles — on the left-hand side of the equation; then solve for "x" ; then substitute that solved value for "x" into the given expressions for BOTH "m∡1" AND "m∡2" ; to find the values for " m∡1" AND " m∡2 " ; which are the values asked for in this very question ;
___________________________________________

 m∡1 + m∡2 + m∡3 = 180 ;
___________________________________________
8x + (2x + 3) + 30 = 180 ;
___________________________________________
8x + 2x + 3 + 30 = 180 ;
___________________________________________
Combine the "like terms" on the 'left-hand side" of the equation; to simplify:
____________________________________________________________
+8x + 2x = +10x ;
___________________________________________
+3 + 30 = +33 ;
___________________________________________
Rewrite the entire equation, as:
___________________________________________
10x + 33 = 180 ;
___________________________________________
Now, subtract "33" from EACH SIDE of the equation:
___________________________________________
10x + 33 − 33 = 180 − 33 ;
___________________________________________
to get:
___________________________________________
10x = 147 ;
____________________________________________
Now, divide EACH side of the equation by "10" ; to isolate "x" on ONE SIDE of the equation; and to solve for "x" :
____________________________________________
10x / 10 = 147 / 10 ;
____________________________________________
to get:
____________________________________________
x = 14.7 ;
___________________________________________
Now, given the following, we plug in our solved value, "14.7", for "x", into the expression given for "m∡1" and "m∡2"; as follows:
________________________________________________

Angle 1: "(8x)" = 8*(14.7) = 117.6° ;
________________________________________________
Angle 2: "2x + 3" = 2*(14.7) + 3 = 29.4 + 3 = 32.4° ;
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These are the two answers; that is the 2 (TWO) values asked for in the question: 117.6° ; 32.4° .
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Do they make sense? That is, do the measurements of ALL 3 (THREE) angles; that is, our two solved measurements added together, and then added to the value of the third angle (given: "m∡3 = 30°); all add up to 180° ?
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Let us check:
_______________________________
m∡1 + m∡2 + m∡3 = 180 ;
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Plugging in our solved values for "m∡1" and "m∡2" ; and our given value: "30" — for "m∡3 — does the equation hold true?
______________________________________________
→ 117.6 + 32.4 + 30 = ? 180 ??
______________________________________________
→ 117.6 + 32.4 = 150 ; → 150 + 30 =? 180 ? Yes!
______________________________________________

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