The value of d is -2
Step-by-step explanation:
The nth term of the arithmetic sequence is [tex]u_{n}=u_{1}+(n-1)d[/tex] , where
- [tex]u_{1}[/tex] is the first term
- d is the common difference between each 2 consecutive terms
The nth term of the geometric sequence is [tex]u_{n}=u_{1}r^{n-1}[/tex] , where
- [tex]u_{1}[/tex] is the first term
- r is the common ratio between each two consecutive terms [tex]r=\frac{u_{2}}{u_{1}}=\frac{u_{3}}{u_{2}}[/tex]
∵ [tex]u_{1}[/tex] of an arithmetic sequence is 1
∵ The common difference is d, where d ≠ 0
∵ [tex]u_{2}[/tex] , [tex]u_{3}[/tex] , [tex]u_{6}[/tex] are the first 3 terms of a geometric sequence
∵ [tex]u_{2}[/tex] = 1 + (2 - 1)d
∴ [tex]u_{2}[/tex] = 1 + d
∵ [tex]u_{3}[/tex] = 1 + (3 - 1)d
∴ [tex]u_{2}[/tex] = 1 + 2d
∵ [tex]u_{6}[/tex] = 1 + (6 - 1)d
∴ [tex]u_{2}[/tex] = 1 + 5d
∴ The first 3 terms of the geometric sequence are (1 + d) , (1 + 2d) ,
(1 + 5d)
∵ The common ratio in the geometric sequence is [tex]r=\frac{u_{2}}{u_{1}}=\frac{u_{3}}{u_{2}}[/tex]
∴ [tex]r=\frac{1+2d}{1+d}=\frac{1+5d}{1+2d}[/tex]
- Use cross multiplication with the equal fractions
∵ [tex]\frac{1+2d}{1+d}=\frac{1+5d}{1+2d}[/tex]
∴ (1 + 2d)(1 + 2d) = (1 + d)(1 + 5d)
∴ 1 + 2d + 2d + 4d² = 1 + 5d + d + 5d²
- Add like terms in each side
∴ 1 + 4d + 4d² = 1 + 6d + 5d²
- Subtract 1 from both sides
∴ 4d + 4d² = 6d + 5d²
- Subtract 4d from both sides
∴ 4d² = 2d + 5d²
- Subtract 4d² from each side
∴ 0 = 2d + d²
- Take d as a common factor
∴ 0 = d(2 + d)
- Equate each factor by 0
∴ d = 0 but we will reject it because d ≠ 0
∴ 2 + d = 0
- Subtract 2 from both sides
∴ d = -2
The value of d is -2
Learn more:
You can learn more about the sequence in brainly.com/question/1522572
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