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anemone
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Find distinct positive integers \(\displaystyle a,\;b,\) and \(\displaystyle c\) such that \(\displaystyle a+b+c,\;ab+bc+ac,\;abc\) forms an arithmetic progression.
Good catch: I completely missed that one.Wilmer said:3,7,16 : 26,181,336 ; diff=155
Not allowed! Those numbers are not distinct.Wilmer said:4,4,24 : 32,208,384 ; diff=176
True...but that was a BONUS: I DID state: (keeping it a =< b =< c)(Ninja)Opalg said:Not allowed! Those numbers are not distinct.
Opalg said:"Write $\sum a$ for $a+b+c$ and $\sum bc$ for $bc+ca+ab$. Then $\sum a,\ \sum bc$ and $abc$ form an arithmetic progression, and therefore $\sum a - 2\sum bc + abc = 0.$ Hence $$\textstyle (a-2)(b-2)(c-2) = abc - 2\sum bc + 4 \sum a - 8 = 3\sum a - 8.$$
That equation tells you quite a lot. For a start, the product $(a-2)(b-2)(c-2)$ is congruent to $1\pmod3$, which means that either $a$, $b$ and $c$ are all multiples of $3$, or one of them is a multiple of $3$ and the other two are congruent to $1\pmod3$. Next, the product of the three numbers $a-2$, $b-2$, $c-2$ is only about three times their sum, so at least one of them must be rather small. You can easily rule out the possibility that the smallest of the numbers $a,\ b,\ c$ is $1$. So that makes it seem almost sure to be $3$. If you then put $a=3$ in the displayed equation",
An Arithmetic Progression (AP) is a sequence of numbers in which each term is obtained by adding a fixed number, called the common difference, to the previous term. The common difference can be positive, negative, or zero. For example, 2, 5, 8, 11, 14, 17, ... is an AP with a common difference of 3.
The formula for the nth term (an) of an AP is an = a1 + (n-1)d, where a1 is the first term and d is the common difference. This formula can be used to find any term in the sequence.
The sum of an AP can be calculated using the formula Sn = n/2[2a1 + (n-1)d], where Sn is the sum of the first n terms of the AP, a1 is the first term, and d is the common difference. This formula is based on the number of terms (n) and the first and last term of the AP.
An Arithmetic Progression is a sequence of numbers in which each term is obtained by adding a fixed number to the previous term, while a Geometric Progression is a sequence of numbers in which each term is obtained by multiplying the previous term by a fixed number, called the common ratio. In an AP, the difference between consecutive terms is constant, while in a GP, the ratio between consecutive terms is constant.
Arithmetic Progressions have several applications in real life, including calculating interest on loans, forecasting stock market trends, and creating schedules for recurring events. They are also used in physics and engineering to model linear relationships and in computer science for data structures and algorithms.