Find the value of k if k-3, 2k-3 and 4k+3 are in GP | Geometric Progression

Find the value of k if k-3, 2k-3 and 4k+3 are in GP | Geometric Progression

In mathematics, a geometric progression, also known as a geometric sequence, is a sequence of non-zero numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. For example, the sequence 2, 6, 18, 54, ... is a geometric progression with common ratio 3. Similarly 10, 5, 2.5, 1.25, ... is a geometric sequence with common ratio 1/2.

Generally, to check whether a given sequence is geometric, one simply checks whether successive entries in the sequence all have the same ratio.

The common ratio of a geometric sequence may be negative, resulting in an alternating sequence, with numbers alternating between positive and negative. For instance

1, −3, 9, −27, 81, −243, ...

is a geometric sequence with common ratio −3.

The behaviour of a geometric sequence depends on the value of the common ratio.
If the common ratio is:
✓positive, the terms will all be the same sign as the initial term.

✓negative, the terms will alternate between positive and negative.

✓greater than 1, there will be exponential growth towards positive or negative infinity (depending on the sign of the initial term).

✓1, the progression is a constant sequence.

✓between −1 and 1 but not zero, there will be exponential decay towards zero (→ 0).

✓−1, the absolute value of each term in the sequence is constant and terms alternate in sign.

✓less than −1, for the absolute values there is exponential growth towards infinity, due to the alternating sign.

✓Geometric sequences (with common ratio not equal to −1, 1 or 0) show exponential growth or exponential decay, as opposed to the linear growth (or decline) of an arithmetic progression such as 4, 15, 26, 37, 48, … (with common difference 11). This result was taken by T.R. Malthus as the mathematical foundation of his Principle of Population. Note that the two kinds of progression are related: exponentiating each term of an arithmetic progression yields a geometric progression, while taking the logarithm of each term in a geometric progression with a positive common ratio yields an arithmetic progression.

[shrawanchouhan]

Mam answer 3 ayaga sayad confusion ho raha hai