If the first term of a G.P. a_1, a_2, a_3, do | vedclass

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(A) Given that the first term $a_1 = 1$ and the common ratio is $r$.Thus,the terms are $a_1 = 1$,$a_2 = r$,and $a_3 = r^2$.We want to minimize the exp

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$-\frac{2}{5}$

Vedclass · Answer

(A) Given that the first term $a_1 = 1$ and the common ratio is $r$.Thus,the terms are $a_1 = 1$,$a_2 = r$,and $a_3 = r^2$.We want to minimize the expression $f(r) = 4a_2 + 5a_3 = 4r + 5r^2$.To find the minimum,we take the derivative with respect to $r$ and set it to zero:$f'(r) = \frac{d}{dr}(4r + 5r^2) = 4 + 10r$.Setting $f'(r) = 0$ gives $4 + 10r = 0$,which implies $10r = -4$.Therefore,$r = -\frac{4}{10} = -\frac{2}{5}$.Since the second derivative $f''(r) = 10 > 0$,the function has a minimum at $r = -\frac{2}{5}$.

If the first term of a $G.P.$ $a_1, a_2, a_3, \dots$ is unity such that $4a_2 + 5a_3$ is least,then the common ratio of the $G.P.$ is

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