a. The next term in the sequence is 303.75.
b. The sequence is a geometric sequence.
c. The recursive function for the sequence is: aₙ = aₙ₋₁ (1.5) for n≥2.
Given the sequence is : 60,90,135,202.5,....
a. the next term in the sequence is:
Find the common ratio by dividing any term in the sequence by the term that comes before it:
a₂/a₁ = 90/60 = 1.5
a₃/a₂ = 135/90 = 1.5
a₄/a₃ = 202.5/135 = 1.5
The common ratio (r) of the sequence is constant and equals the quotient of two consecutive terms.
r = 1.5
find the sum:
To find the sum of the series, plug the first term: a = 60, the common ratio(r): 1.5 and the number of elements n = 4 into the geometric series sum formula:
Sₙ = a(1₋rⁿ/1₋r)
Sₙ = 60(1 ₋ 1.5⁴/1₋1.5)
Sₙ = 60(1 ₋ 5.0625/1₋1.5)
S₄ = 60(8.125)
S₄ = 478.5
To find the general form of the series, plug the first term:
aₙ = 60×1.5ⁿ⁻¹
use the general form to find the next term:
a₁ = 60
a₂ = a₁rⁿ⁻¹ = 60(1.5)¹ = 90
a₃ = a₁rⁿ⁻¹ = 60(1.5)² = 135
a₄ = a₁rⁿ⁻¹ = 60(1.5)³ = 202.5
a₄ = a₁rⁿ⁻¹ = 60(1.5)⁴ = 303.75
hence the required next term is 303.75
(b) The above sequence is geometric sequence.
(c) recursive function for the sequence : aₙ = aₙ₋₁ (1.5) for n≥2.
Hence we get the required answers.
Learn more about Geometric progressions here:
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