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I have a sum that has the following form.

$\underbrace{\underbrace{(7 + 2*1)}_\text{A} + {(7 + 2*A)}}_\text{B} + (7 + 2 * B) ....$

The first term is calculated for $n=1$, the second term for $n=2$ and so on. Each parantheses include the result of the previously caluclated term, as shown above. I am trying to find a formula, that lets you calculate The sum for any $n$.

kklaw
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  • Your question isn't clear at all. I suggest to elaborate your answer and show your attempts as well. It's how the community receives your question. – RAHUL Nov 09 '21 at 17:15

2 Answers2

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The sum you mention follows the recursive law $$ x_1 = 9, \quad x_{n+1} = x_n + (7+2x_n) $$

This is a linear difference equation that you can solve to get $$ x_n = \underbrace{\frac{25}{6} \cdot 3^n}_{y_n} \,\,\underbrace{-\,\,\frac 72}_{y_n^*} $$

We start by solving the homogeneous equation $y_{n+1}-3 y_n = 0$, that gives you, by simple recursion, $y_n = c\cdot 3^n$. Then, the solution of the non-homogeneous equation is $x_n = y_n + y^*_n$, where $y^*_n$ is a particular solution. Trying $y^*_n = k$ and substituting back in the equation, you get $y^*_n = -\frac 72$. Finally the constant $c$ is computed by requiring that $x_1=9$.

PierreCarre
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This is a recursive definition.

$a_1=7+2\cdot 1$

$a_2=a_1+(7+2\cdot a_1)$

$a_3=a_2+(7+2\cdot a_2)$

$a_{i+1}=a_i+(7+2a_i)=7+3a_i$

Can be one way. You could also use the $\sum$-notation.

As you have no "abstract" values (that depend on an undetermined $x$ or something), you can also just calculate the values, and try to give a closed formula.

Cornman
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  • Yeah, I would actually prefer a closed formula, so I can calculate the value for any $n$. – kklaw Nov 09 '21 at 17:19
  • Then calculate $a_1, a_2, a_3$ and so on. Try to guess the formula, and prove by induction that it holds. – Cornman Nov 09 '21 at 17:21
  • @F.V. One can derive a closed form for $a_i$ based on the recurrence $a_{i+1} = 7 + 3a_i$. Do you know how to do so? Based on that, we can derive a closed form for $ \sum a_i$. – Calvin Lin Nov 09 '21 at 17:21
  • Unfortunately I do not, is there a technique for it? – kklaw Nov 09 '21 at 17:22