an arithmetic sequence u1 u2 u3 ...has u1 = 1 and common difference d is not = 0 given that u2 u3 and u6 are the first three terms if a geometric sequence find the value of d
Answers
Solution:
- u1, u2, u3, ... are in A.P.
Here, a = 1 and d ≠ 0
- u2, u3 and u6 are in G.P.
i.e. u2 = 1 + d, u3 = 1 + 2d and u6 = 1 + 5d are in G.P.
Therefore, (1 + 2d)² = (1 + d) (1 + 5d)
» 1 + 4d² + 4d = 1 + 6d + 5d²
» d² + 2d = 0
» d (d + 2) = 0
Since, d ≠ 0, d = - 2
Therefore, common difference is - 2.
Given : An arithmetic sequence u₁, u₂, u₃ ... has u₁ = 1 and common difference d is not = 0
u₂, u₃ and u₆ are the first three terms of a geometric sequence
To Find: the value of d
Solution:
Arithmetic sequence
Sequence of terms in which difference between one term and the next is a constant.
This is also called Arithmetic Progression AP
Arithmetic sequence can be represented in the form :
a, a + d , a + 2d , …………………………, a + (n-1)d
a = First term
d = common difference = aₙ-aₙ₋₁
nth term = aₙ = a + (n-1)d
Geometric sequence
A sequence of numbers in which the ratio between consecutive terms is constant and called the common ratio.
a , ar , ar² , ... , arⁿ⁻¹
The nth term of a geometric sequence with the first term a and the common ratio r is given by: aₙ = arⁿ⁻¹
u₁, u₂, u₃ ... has u₁ = 1 and common difference d
u₂ = 1 + d
u₃ = 1 + 2d
u₆ = 1 + 5d
These are first three terms of a geometric sequence
=> (1 + 2d)² = (1 + d)(1 + 5d)
=> 1 + 4d² + 4d = 1 + 6d + 5d²
=> d² + 2d = 0
=> d(d + 2) = 0
=> d = 0 , d = - 2
but d ≠ 0
Hence d = - 2
Value of d is - 2
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