Question:
The geometric sequence ai is defined by the formula: a₁ = 8, aᵢ = aᵢ₋₁(-1.5 ).
Find the sum of the first 20 terms in the sequence. Choose 1 answer:
Answer:
The sun of 20 terms of the progress is -10,637.621536256
Step-by-step explanation:
Given
Geometric Sequence
a₁ = 8
aᵢ = aᵢ₋₁(-1.5 )
First, the common ratio needs to be calculated.
The common ratio is the ratio of a term to its previous term.
In other words,
Ratio = 2nd term ÷ 1st term or 3rd term ÷ 2nd term, ...... Etc.
We can calculate the common ratio from aᵢ = aᵢ₋₁(-1.5 ) by dividing both sides by aᵢ₋₁. This gives
aᵢ / aᵢ₋₁ = aᵢ₋₁(-1.5 ) / aᵢ₋₁
aᵢ / aᵢ₋₁ = -1.5
So, the common ratio, r = -1.5
Now that we've had the common ratio and first term to be -1.5 and 8 respectively, the sum of 20 terms can then be calculated using the sum of n terms formula.
Sₙ = a(1 - rⁿ)/(1 - r)
We're making use of this formula because r is less than 1
Where n = 20
a = first term = 8
r = -1.5
By substituting these values; we get
S₂₀ = 8(1 - (-1.5)²⁰)/(1 - (-1.5))
S₂₀ = 8(1 - (-1.5)²⁰)/(1 + 1.5))
S₂₀ = 8(1 - (-1.5)²⁰)/(1 + 1.5))
S₂₀ = 8(1 - (3325.25673008
))/(2.5)
S₂₀ = 8(1 - 3325.25673008
)/(2.5)
S₂₀ = 8(-3324.25673008
)/(2.5)
S₂₀ = -10,637.621536256
