- #1
ramsey2879
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Given [tex]a_{0} = 0[/tex], [tex]a_{1} = 1[/tex], and [tex]a_n = Ba_{n-1} - a_{n-2}[/tex]; prove that
[tex]\sum_{i=1}^{m} a_{2i-1} = (a_{m})^2[/tex]
[tex]\sum_{i=1}^{m} a_{2i-1} = (a_{m})^2[/tex]
ramsey2879 said:Given [tex]a_{0} = 0[/tex], [tex]a_{1} = 1[/tex], and [tex]a_n = Ba_{n-1} - a_{n-2}[/tex]; prove that
[tex]\sum_{i=1}^{m} a_{2i-1} = (a_{m})^2[/tex]
Good point. At least it helps in the summing of the other half of the series. I will edit my other post to complete my proof now so I will not need this relation at this moment.tiny-tim said:Hi ramsey
[tex]\sum_{i=1}^{m} a_{2i-1}\,=\,\sum_{i=1}^{m} Ba_{2i-2}\,-\,\sum_{i=0}^{m-1} a_{2i-1}[/tex]
Does that help?
The "Prove Sum of Series Property" is a mathematical concept that states the sum of a series of numbers can be calculated by using the formula a_n = Ba_{n-1} - a_{n-2}, where a_n is the nth term in the series, B is a constant, and a_{n-1} and a_{n-2} are the previous two terms in the series.
The "Prove Sum of Series Property" is used to simplify the calculation of the sum of a series of numbers. Instead of adding each term individually, this formula can be used to find the sum more efficiently.
Yes, the "Prove Sum of Series Property" can be applied to all series as long as the series follows the pattern a_n = Ba_{n-1} - a_{n-2}. This includes arithmetic, geometric, and other types of series.
The "Prove Sum of Series Property" can be proved by using mathematical induction, which involves showing that the formula holds for the first few terms in the series and then proving that it holds for all subsequent terms. This is a commonly used method in mathematical proofs.
Yes, the "Prove Sum of Series Property" can be generalized to higher orders by using the formula a_n = Ba_{n-1} - a_{n-2} + c_{n-3} - ... + (-1)^{n-1}c_{1} + (-1)^{n}c_{0}, where c_{i} are constants. This formula is known as the "Sum of Finite Series Property" and can be used to calculate the sum of any finite series.