
The pattern having the fastest growth rate is :
It is a geometric sequence. The next two terms are: 0.125, 0.0625
The given sequence is:
2, 1, 0.5, 0.25, . . . .
Let a(n) be the nth term, then in the above sequence
a(1) = 2
a(2) = 1
a(3) = 0.5
a(4) = 0.25
Notice that:
a(2) = a(1) x 0.5
a(3) = a(2) x 0.5
a(4) = a(3) x 0.5
A sequence, of which its next term is obtained by multiplying the previous term by a constant is called a geometric sequence. The constant is called the ratio.
Therefore this is a geometric sequence with r = 0.5
To find the next two terms, continue the multiplication process:
a(5) = a(4) x 0.5 = 0.25 x 0.5 = 0.125
a(6) = a(5) x 0.5 = 0.125 x 0.5 = 0.0625
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The function increases in the interval (-∞, -3) and the function also increases in the interval (-1,∞) .
The given function is of the form
f(x) = x³ + 6x² + 8x
Now we take the first differentiation of the function
f'(x) = 2x² + 12x + 8
f'(x) = 2 (x² + 6x + 9) -10
f'(x) = 2(x+3)² - 10
Therefore at x = -3 , f'(x) = -10.
Hence the function is increasing in the interval of (-∞, -3)
Again f'(x) = 2x² + 12x + 8 , so after first differentiation we get :
That the function is also increasing in the interval (-1,∞)
Now for the interval (-4,-2), we can say that the graph of the function is positive as the y value increases and then decreases but all y values are positive as illustrated in the graph.
In the interval (0,∞) the function is strictly increasing and has positive values only.
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