Let [tex]a+b,a-b,ab,\frac{a}{b}[/tex] form an arithmetic sequence. What is the sixth term of this sequence?
Please answer with solution, thanks in advance! :))
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Let [tex]a+b,a-b,ab,\frac{a}{b}[/tex] form an arithmetic sequence. What is the sixth term of this sequence?
Please answer with solution, thanks in advance! :))
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Answer:
a + b; a - b; ab; a/b.
Here, we need to find the values of a and b.
First, if there are terms x, y and z in a progression, then 2y = z + x.
Therefore,
2(a - b) = a + b + ab.
Simplifying this further, we get:
a - 3b = ab.
Now, again look at the 2nd, 3rd and 4th term. They are also obviously in an AP.
2ab = a - b + a/b.
substitute ab = a - 3b.
2(a - 3b) = a - b + a/b.
2a - 6b = a - b + a/b.
a - 5b = a/b.
ab - 5b² = a.
Again, substitute a = ab + 3b.
ab - 5b² = ab + 3b.
-5b² = 3b
-5b = 3
b = -3/5 (yay!)
Substitute b in any equation.
a = a(-3/5) + 3(-3/5)
a + 3a/5 = 9/5(-1)
-(5a + 3a)/5 = 9/5
-8a/5 = 9/5
a = -9/8.
Now,
Now, this is an AP.
First term = -1.725
Difference = 1.2
Number of terms = 6.
a₆ = -1.725+(6-1)(1.2)
=> 4.275
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