**Definition of algebra:**

Algebra is that branch of mathematics in which parameters are represented by variables, and the problems and expression are indicated by symbols. The algebraic concept of notation is to abridge and common the analysis of the mathematical problems. So algebra referenced as the universal arithmetic. Below the problems are help to improve the algebra solving skills

A geometrical algebra progression is a series of parameters, each of which is multiplied by the equal constant factor. These constant factor is known as ratio ; then the first equation is contain positive variables, the geometrical progression shall be an increasing, or a decreasing series, depend upon progression the ratio is greater or less, than unity.

Thus, 4, 8, 24, 72, 216,.........

is an increase the geometrical series , in which the ratio is 3 ; and

243,81, 27, 9, 3, 1, 1/3,.........

**Solving Formulas:**

The two primitive equations,

** k=cq ^{m}**

**R= c(q ^{m}-1) / q -1** , -----------(1)

contain solving the five quantities, c,q, l, m, R, any three of which being

**k=l/q ^{m-1}** -------------(2)

**Example 1:**

The first term of a geometrical solving progression is 3, and the ratio is 2; find the 12^{th} term and the sum of the series.

**Solution:**

we have given

c=3, q = 2, m =12.

When by formulas ,

**k=cq ^{m-1}**

k= 3*2^{12} = 3x 2048 = 6144.

**R=c(q ^{m}-1)/ q-1**

R=3(2^{12}-1)/(2-1)= 3* 4095 =12285

**Answer is:** 6144 , 12285.

**Example 2:**

The sum of a geometrical solving progression is 1820, the number of terms 6, and the ratio 3; find the first terms , and the last terms.

**solution: **

We have given,

R= 1820, m = 6, q = 3.

By formula,

**R=c(q ^{m}-1/(q-1),**

1820 = a(3^{6}-1) / (3-1) = 364c ;

c = 5 ; first term.

Then by formula ,

k = 5*3^{6} = 1215, last term.

**Example 3:**

It is required to find 3 geometrical means between 6 and 436.

**Solution:**

By formula : **q = mk+1 √ k/c**

q = 4√ 486/ 6 = 4 √81 = 3.

Therefore, the series is 6, 18, 54, 162, 484.

**Answer is**:6, 18, 54, 162, 484.