A geometric sequence , the fifth term is four times the third terms, the second term is 4.
If r<0, determine:
A) the value of r, the common ratio.
B) the value of a.
C) the tenth term.
D) the sum of the 15 terms.
You have a GP where 2nd term is 4 -----> ar=4
the fifth term is four times the third term ---- ar^4 = ar^2
ar^4 = ar^2
divide by ar^2, r can't be zero
r^2 = 1
r = ± 1
then in ar=4
if r = +1, a = 4
if r = -1, a = -4
Do C and D by simply using the formulas that you learned.
In geometric sequence:
an = a1 ∙ r ⁿ ⁻ ¹
a2 = a1 ∙ r
a3 = a1 ∙ r ²
a5 = a1 ∙ r ⁴
The fifth term is four times the third terms mean; a5 = 4 ∙ a3
a1 ∙ r ⁴ = 4 ∙ a1 ∙ r ²
Divide both sides by ( a1 ∙ r ² )
r ² = 4
r = ± √4
r = ± 2
In this case r < 0 so:
r = - 2
The second term is 4 mean; a2 = 4
a1 ∙ r = 4
a1 ∙ ( - 2 ) = 4
Divide both sides by - 2
a1 = 4 / - 2
a1 = - 2
a10 = a1 ∙ r⁹
a10 = ( - 2 ) ∙ ( - 2 )⁹
a10 = ( - 2 ) ∙ ( - 512 )
a10 = 1024
Sn = a1 ( 1 − r ⁿ ) / ( 1 − r )
In this case n = 10
S10 = ( - 2 ) ∙ [ 1 − ( - 2 )¹⁰ ) / [ 1 − ( - 2 ) ]
S10 = ( - 2 ) ∙ ( 1 − 1024 ) / ( 1 + 2 )
S10 = ( - 2 ) ∙ ( − 1023 ) / 3
S10 = 2046 / 3
S10 = 682
The sum of the 15 terms.
Sn = a1 ( 1 − r ⁿ ) / ( 1 − r )
where n = 15
S15 = ( - 2 ) ∙ [ 1 − ( - 2 )¹⁵ ] / [ 1 − ( - 2 ) ]
S15 = ( - 2 ) ∙ [ 1 − ( - 32 768 ) ] / ( 1 + 2 )
S15 = ( - 2 ) ∙ ( 1 + 32 768 ) / 3
S15 = ( - 2 ) ∙ 32 769 / 3
S15 = - 65 538 / 3
S15 = - 21 846
To find the answers to each part of the question, we'll need to apply the properties of a geometric sequence and use the given information.
Let's start with part A) finding the value of r, the common ratio.
We know that the second term of the sequence is 4, so we can write the sequence as: a, 4, ar, ar^2, ar^3, ...
Since the fifth term is four times the third term, we have:
ar^3 = 4(ar)
r^2 = 4
Since we are given that r < 0, we take the negative square root to get r = -2.
Moving on to part B) finding the value of a.
We are given that the second term is 4, so the sequence becomes: a, 4, -2a, -4a, -8a, ...
From this, we can see that a = 4, as it is the value corresponding to the first term.
Now for part C) finding the tenth term.
We know that the first term, a = 4, and the common ratio, r = -2. Using these values, we can find the tenth term:
Tenth term = a * r^(n-1)
Tenth term = 4 * (-2)^(10-1)
Tenth term = 4 * (-2)^9
Tenth term = 4 * (-512)
Tenth term = -2048
Therefore, the tenth term is -2048.
Finally, for part D) finding the sum of the 15 terms.
The sum of a finite geometric series is given by the formula:
Sum = a * (1 - r^n) / (1 - r)
In this case, we need to find the sum of the first 15 terms. Therefore, n = 15.
Sum = 4 * (1 - (-2)^15) / (1 - (-2))
Sum = 4 * (1 - 32768) / (1 + 2)
Sum = 4 * (-32767) / 3
Sum = -43689.33...
Therefore, the sum of the first 15 terms is approximately -43689.33.