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If an infinite GP has the first term x and the sum 3, then which one of the following is correct?
1. x < -6
2. -6 < x < 0
3. 0 < x < 6
4. None of these

1 Answer

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Best answer
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

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