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Common Ratio

The first ter...

Question

The first term of the geometric progression $b_{1}, b_{2}, b_{3}, . . .$ is unity. For what value of the common ratio of the progression is $4 b_{2} + 5 b_{3}$ at a minimum?

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Solution

given $b_{1} = 1$
$⟹ 4 b_{2} + 5 b_{3} = 4 r + 5 r^{2}$ where r is common ratio
A quadratic equation has a minimum value at x=-b/2a $⟹ 4 b_{2} + 5 b_{3}$ has minimum value at $r = - 0.4$

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b_{1}, b_{2}, b_{3}, \dots

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4 b_{2} + 5 b_{3}

b_{1}, b_{2}, b_{3}, \dots

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4 b_{2} + 5 b_{3}

a, a_{1}, a_{2}, a_{3}, . . . . . . . . .

b, b_{1}, b_{2}, b_{3}, . . . . . . . . .

l o g a_{n} - b_{n} = l o g a - b

β

a

r

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2 r

3 S

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