Find the sum up to n terms: 1+11+111+1111...... |Geometric Progression Sum of first n terms of GP
Find the sum up to n terms: 1+11+111+1111...... | Geometric Progression Sum of first n terms of GP 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. Facebook: https://www.facebook.com/Rashmi-Rekha-Dehingia-105105207917015/ Instagram: https://www.instagram.com/rashmirekhadehingia/ Telegram: https://t.me/rashmi_mathemaics #rashmirekhadehingia #geometricprogresion
Find the sum up to n terms: 1+11+111+1111...... | Geometric Progression Sum of first n terms of GP 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. Facebook: https://www.facebook.com/Rashmi-Rekha-Dehingia-105105207917015/ Instagram: https://www.instagram.com/rashmirekhadehingia/ Telegram: https://t.me/rashmi_mathemaics #rashmirekhadehingia #geometricprogresion