Money accumulates in a fund at an annual effective interest rate of i during the first $5$ years, and at an annual effective interest rate of $2$i thereafter.

A deposit of $1$ is made into the fund at time $0.$ It accumulates to $3\mathrm{.}09$ at the end of $10$ years and to $13\mathrm{.}62$ at the end of $20$ years. What is the value of the deposit at the end of $7$ years?

$($This question is worth $2$ points$)$

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a $,1\mathrm{.}90$

b $,1\mathrm{.}98$

$2\mathrm{.}13$

d $2\mathrm{.}56$