Respuesta :
The twentieth term of the arithmetic and the geometric sequence will be 7.333 and 235.38.
How to calculate the sequence?
From the information given, the arithmetic and geometric sequence both have first term 1, and their second term are equal. This will be:
a + d = ar
where, a = 1
1 + d = r
d = r - 1
The 14th term of the arithmetic sequence is three times the third term of the geometric sequence. This will be:
a + 13d = 3(ar²)
1 + 13(r - 1) = 3(ar²)
1 + 13r - 13 = 3ar²
13r - 12 = 3r²
3r² - 13r + 12 = 0
3r² - 9r - 4r + 12 = 0
3r(r - 3) - 4(r - 3) = 0
(3r - 4) = 0.
3r = 4
r = 4/3
Therefore, common ratio is 4/3.
The common difference (d) will be:
d = r - 1
d = 4/3 - 1
d = 1/3
The twentieth term of the arithmetic sequence will be:
= a + 19d
= 1 + (19 × 1/3)
= 1 + 6 1/3
= 7 1/3 = 7.33
The twentieth term of the geometric sequence will be:
= ar^19
= (4/3)^19
= 235.38
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