We will discuss about some of the properties of Geometric Progressions and geometric series which we will frequently use in solving different types of problems on Geometric Progressions.
Property I: When each term of a Geometric Progression is multiplied or divided by a same non-zero quantity, then the new series forms a Geometric Progression having the same common ratio.
Proof:
Let, a\(_{1}\), a\(_{2}\), a\(_{3}\), a\(_{4}\), .................., a\(_{n}\), .......... be a Geometric Progression with common r. Then,
\(\frac{a_{n + 1}}{a_{n}}\) = r, for all n ∈ N ................... (i)
Let k be a non-zero constant. Multiplying all the terms of the
given Geometric Progression by k, we obtain the sequence
ka\(_{1}\), ka\(_{2}\), ka\(_{3}\), ka\(_{4}\), ................., ka\(_{n}\), ................
Clearly, \(\frac{ka_{(n + 1)}}{ka_{n}}\) = \(\frac{a_{(n + 1)}}{a_{n}}\) = r for all n ∈ N [Using (i)]
Hence, the new sequence also forms a Geometric Progression with common ratio r.
Property II: In a Geometric Progression the reciprocals of the terms also form a Geometric Progression.
Proof:
Let, a\(_{1}\), a\(_{2}\), a\(_{3}\), a\(_{4}\), .................., a\(_{n}\), .......... be a Geometric Progression with common r. Then,
\(\frac{a_{n + 1}}{a_{n}}\) = r, for all n ∈ N ................... (i)
The series formed by the reciprocals of the terms of the given Geometric Progression is
\(\frac{1}{a_{1}}\), \(\frac{1}{a_{2}}\), \(\frac{1}{a_{3}}\), ................., \(\frac{1}{a_{n}}\), .................
We have, \(\frac{\frac{1}{a_(n + 1)}}{\frac{1}{a_{n}}}\) = \(\frac{a_{n}}{a_{n + 1}}\) = \(\frac{1}{r}\) [Using (i)]
So, the new series is a Geometric Progression with common ratio \(\frac{1}{r}\).
Property III: When all the terms of a Geometric Progression be raised to the same power, then the new series also forms a Geometric Progression.
Proof:
Let, a\(_{1}\), a\(_{2}\), a\(_{3}\), a\(_{4}\), .................., a\(_{n}\), .......... be a Geometric Progression with common r. Then,
a_(n + 1)/a_n = r, for all n ∈ N ................... (i)
Let k be a non-zero real number. Consider the sequence
a1^k, a2^k, a3^k, ........, an^k, ...........
We have, a_(n +1)^k/a_n^k = (a_(n +1)/a_n)^k = r^k for all n ∈ N, [Using (i)]
Hence, a1^k, a2^k, a3^k, ........, an^k, ........... is a Geometric Progression with common ratio r^k.
Property IV: The product of the first and the last term is always equal to the product of the terms equidistant from the beginning and the end of finite Geometric Progression.
Proof:
Let, a\(_{1}\), a\(_{2}\), a\(_{3}\), a\(_{4}\), .................., a\(_{n}\), .......... be a Geometric Progression with common r. Then,
Kth term form the beginning = a_k = a_1r^(k - 1)
Kth term from the end = (n – k + 1)th term form the beginning
= a_(n – k + 1) = a_1r^(n – k)
Therefore, kth term from the beginning)(kth term from the end) = a_ka_(n – k + 1)
= a1r^(k – 1)a1r^(n – k) = a162 r^(n -1) = a1 * a1r^(n – 1) = a1an for all k = 2, 3, ......, n - 1.
Hence, the product of the terms equidistant from the beginning and the end is always same and is equal to the product of the first and the last term.
Property V: Three non-zero quantity a, b, c are in Geometric Progression if and only if b^2 = ac.
Proof:
A, b, c are in Geometric Progression ⇔ b/a = c/b = common ratio ⇔ b^2 = ac
Note: When a, b, c are in Geometric Progression, then b is known as the geometric mean of a and c.
Property VI: When the terms of a Geometric Progression are selected at intervals then the new series obtained also a Geometric Progression.
Property VII: In a Geometric Progression of non-zero non-negative terms, then logarithm of each term is form an Arithmetic Progression and vice-versa.
i.e., If a\(_{1}\), a\(_{2}\), a\(_{3}\), a\(_{4}\), .................., a\(_{n}\), ..................... are non-zero non-negative terms of a Geometric Progression then loga1, loga2, loga3, loga4, ....................., logan, ......................... forms an Arithmetic Progression and vice-versa.
Proof:
If a\(_{1}\), a\(_{2}\), a\(_{3}\), a\(_{4}\), .................., a\(_{n}\), ..................... is a Geometric Progression of non-zero non-negative terms with common ratio r. Then,
a_n = a1r^(n -1), for all n ∈ N
⇒ log a_n = log a1 + (n – 1) log r, for all n ∈ N
Let b_n = log a_n = log a1 + (n – 1) log r, for all n ∈ N
Then, b_ n +1 – b_n = [loga1 + n log r] – [log a1 + (n -1) log r] = log r, for all n ∈ N.
Clearly, b_n + 1 – b_n = log r = constant for all n ∈ N. Hence, b1, b2, b3, b4, ................., bn, ....... i.e., log a1, log a2, log a3, log a4, ..................., log an, ........... be an Arithmetic Progression with common difference log r.
Conversely, let log a1, log a2, log a3, log a4, ..................., log an, ........... be an Arithmetic Progression with common difference d. Then,
log a _(n + 1) – log an = d, for all n ∈ N.
⇒ log (a_n +1/an) = d, for all n ∈ N.
⇒ a_n +1/an = e^d, for all n ∈ N.
⇒ a\(_{1}\), a\(_{2}\), a\(_{3}\), a\(_{4}\), .................., a\(_{n}\), ..................... is a Geometric Progression with common ratio e^d.
● Geometric Progression
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