Terms of geometric progression
Terms of a geometric progression are numbers arranged strictly by ordinal numbers, where the ordinal number itself determines the value of the sequence term. The first term of a geometric progression can be any number except zero (b≠0). In order to find n a term of the geometric progression, it is necessary to multiply the first term by the ratio of the progression the required number of times.
The ratio of the progression is a given number, which remains constant throughout the entire numeric series. To see the essence of the sequence, consider a numeric series where bn- these are the first few terms of the progression with ordinal number n, and q - this is the ratio of the progression.
b1
b2=b1 q
b3=b2 q=b1 qq=b1 q2
b4=b3 q=b1 q2 q=b1 q3
…
From here it is clearly visible that the ratio of the geometric progression is raised to a power, the exponent of which is one less than the ordinal number of the term of the progression that needs to be found, and all terms depend on the first. The general formula for terms of a geometric progression will look like this: bn=b1 q(n-1)
Based on this, knowing the first term of the geometric progression, you can find the first three, four terms of the progression by multiplying by the ratio in the required power. Such an online calculator calculates in reverse, that is, knowing any of the sequence terms, you can find the first. To perform such an operation, the calculator reverses the formula, in which the first term of the geometric progression will be equal to the ratio of the given term to the ratio raised to the power n-1, where n - this is the ordinal number of the known term.
Another way to find the first term of a geometric progression is laid in the definition of the sum of the first few terms of the progression. The sum itself is equal to the product of the first term of the progression and the difference between the ratio raised to the power of the ordinal number of the last participating term and one, then the obtained result needs to be divided by another difference of the ratio, this time without power, and one:
The order of the minuend and subtrahend in the brackets may change, this will not affect the result as long as it happens synchronously:
Then, when redistributing the parameters in the formula, it turns out that the first term of the progression is equal to the product of the sum with the difference of one and the ratio, divided by the difference of one and the ratio in the power of n:
