According to definition, the geometrical progression is the sequence unequal to zero numbers, everyone the subsequent of which equally previous, increased by some constant number (progression denominator). At the same time in a geometrical progression there should not be no zero, otherwise all sequence "will be nullified" that contradicts definition. To find a denominator it is enough to know values of two of her next members. However, not always statements of the problem are so simple.

## It is required to you

- calculator

## Instruction

1. Divide any member of a progression into previous. If the value of the previous member of a progression is unknown or is not defined (for example, for the first member of a progression), then divide value of the subsequent member of a progression on any member of the sequence. As any member of a geometrical progression is not equal to zero, when performing this operation there should not be problems.

2. Example. Let there is a sequence of numbers: 10, 30, 90, 270... It is required to find a denominator of a geometrical progression. Decision: 1 option. Let's take any member of a progression (for example, 90) and we will divide him into previous (30): 90/30=3.2 option. Let's take any member of a geometrical progression (for example, 10) and we will divide into him the subsequent (30): 30/10=3. Answer: denominator of a geometrical progression 10, 30, 90, 270... it is equal to 3.

3. If values of members of a geometrical progression are set not obviously, and in the form of ratios, then work out and solve the system of the equations. Example. The sum of the first and fourth member of a geometrical progression equals 400 (b1+b4=400), and the sum of the second and fifth member equals 100 (b2+b5=100). It is required to find a progression denominator. Decision: Write down a statement of the problem in the form of the system of the equations: b1+b4=400b2+b5=100iz of definition of a geometrical progression follows that: b2=b1*qb4=b1*q^3b5=b1*q^4 where q is the standard designation of a denominator of a geometrical progression. Having substituted in the system of the equations of value of members of a progression, receive: b1 + b1*q^3=400b1*q + decomposition b1*q^4=100После on multipliers turns out: b1 * (1+q^3) =400b1*q(1+q^3) =100teper divide the corresponding parts of the second equation into the first: [b1*q(1+q^3)] / [b1 * (1+q^3)] = 100/400, from where: q=1/4.

4. If the sum of several members of a geometrical progression or the sum of all members of the decreasing geometrical progression is known, then for finding of a denominator of a progression use the corresponding formulas: Sn = b1 * (1-q^n)/(1-q) where Sn is the sum of n of the first members of a geometrical progression and S = b1 / (1-q) where S is the sum of infinitely decreasing geometrical progression (the sum of all members of a progression with a denominator smaller units). Example. The first member of the decreasing geometrical progression is equal to unit, and the sum of all her members is equal to two. It is required to define a denominator of this progression. Decision: Substitute data from a task in a formula. It will turn out: 2=1/(1-q), from where – q=1/2.