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Let \(i\) be the annual effective interest rate and \(n\) and \(p\) be any natural numbers.

i. Define algebraically, in a summation format, \(\ddot{a}_{M_{i}}^{(p)}\).
ii. Prove that \(\ddot{a}_{m+n_{i}}^{(p)}=\ddot{a}_{m_{i}}^{(p)}+v^{m} \ddot{a}_{M_{i}}^{(p)}\)
in Data Science & Statistics by Platinum (119,120 points)
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\ddot{a}_{n}^{(p)}=\sum_{j=0}^{n p-1} \frac{1}{p}(1+i)^{-\frac{j}{p}}=\sum_{j=0}^{n p-1} \frac{1}{p} v^{\frac{j}{p}}=\frac{1}{p}+\frac{1}{p} v^{y_{p}}+\frac{1}{p} v^{\% / p}+\ldots \frac{1}{p} v^{n-\frac{1}{p}}
&\ddot{a}_{n+m}^{(p)}=\frac{1-v^{m+n}}{d^{(p)}} \\
&=\frac{1-v^{m}+v^{m}-v^{m+n}}{d^{(p)}}=\frac{1-v^{m}}{d^{(p)}}+v^{m}\left(\frac{1-v^{n}}{d^{(p)}}\right) \\
&=\ddot{a}_{m}^{(p)}+v^{m} \ddot{a}_{n}^{(p)}
by Platinum (119,120 points)

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