Correct Answer - Option 3 : 0 < x < 6
Concept:
Sum to Infinity of GP
If the number of terms in a GP is not finite, then the GP is called infinite GP. The formula to find the sum to infinity of the given GP is:
\(\rm S_{\infty } = Σ _{n = 1}^{\infty } a(r^{n} - 1) = \frac{a}{1 - r}; -1 < r < 1\)
Here,
S∞ = Sum of infinite geometric progression
a = First term
r = Common ratio
n = Number of terms
Σ(called sigma) means "sum up"
And below and above it is shown the starting and ending values.
Calculation:
We know that the sum of infinite terms of GP is
\(\rm S_{\infty } = \begin{cases} \frac{a}{1 - r}; & \text{ if } x= |r|<1\\ \infty; & \text{ if } x= |r|\geq 1 \end{cases}\)
\(\rm S_{\infty } = \frac{x}{1 - r} = 3, since |r|<1\)
1 - r = \(\rm \frac{x}{3}\)
3 - 3r = x
3 - x = 3r
r = \(\rm \frac{3 - x}{3}\)
-1 < \(\rm \frac{3 - x}{3}\) < 1
-3 < 3 - x < 3
-6 < -x < 0
0 < x < 6
If in a sequence of terms, each succeeding term is generated by multiplying each preceding term with a constant value, then the sequence is called a geometric progression. (GP), whereas the constant value is called the common ratio. For example, 2, 4, 8, 16, 32, 64, … is a GP, where the common ratio is 2.
In General, we write a Geometric Sequence like this:
{a, ar, ar2, ar3, ... }
where:
-
a is the first term, and
-
r is the factor between the terms (called the "common ratio")
Sum of nth terms of G.P.
Consider the G.P,
a,ar,ar2,…..arn−1
Let Sn, a, r be the sum of n terms, first term, and the ratio of the G.P respectively.
Formula:
\(\rm S_{n} = \frac{a(r^{n} - 1)}{r - 1}\), Where r ≠ 1
Properties of Geometric Progression:
The following are the properties of G.P:
- If we multiply or divide a non-zero quantity to each term of the G.P, then the resulting sequence is also in G.P with the same common difference.
- Reciprocal of all the terms in G.P also form a G.P.
- If all the terms in a G.P are raised to the same power, then the new series is also in G.P.
- If y² = xz, then the three non-zero terms x, y, and z are in G.P