Title 18 Geometric Progression.pdf ✅

MSTC 18: GEOMETRIC PROGRESSION 1. Geometric Progression Geometric Progressions are distinguishable by having a common ratio between terms. • Common Ratio – Usually denoted as r • First Term – Usually denoted as a. 2. General Term The first few terms of a geometric progression are a1 = a a2 = ar a3 = ar 2 a4 = ar 3 Notice that in the first few terms, the exponent of r changes, and it follows the rule n − 1. Therefore, the general term of a geometric progression is an = ar n−1 3. Partial Sum Let Sn be the sum of the first n terms of the sequence. The sum Sn = a + ar + ar 2 + ⋯ + ar n−2 + ar n−1 Multiply both sides by r rSn = ar + ar 2 + ⋯ + ar n−1 + ar n Subtract the two equations (1 − r)Sn = a − ar n Solve for Sn Sn = a(1 − r n) 1 − r Therefore, a partial sum formula is Sn = a(1 − r n) 1 − r 4. Infinite Geometric Series Infinite geometric series only converges if and only if |r| < 1. As n → ∞, r n → 0. Therefore, the formula for the sum is S∞ = a 1 − r

5.3. The sum of m geometric means between x and y This formula skips the step of solving for the common ratio. [DERIVATION] General Term an = ar n−1 The first term is x, and the (m + 2)th term is y y = xr m+1 Solve for r r = ( y x ) 1 m+1 Using the formula for the partial sum where the first and last term is known, Sn = a − ran 1 − r The sum of means where the first term is x ( y x ) 1 m+1 and x ( y x ) m m+1 Sm = x ( y x ) 1 m+1 − ( y x ) 1 m+1 x ( y x ) m m+1 1 − ( y x ) 1 m+1 Simplify Sm = y √x m+1 − x √y m+1 √y m+1 − √x m+1 Therefore, the sum of m geometric means in between two numbers is Sm = y √x m+1 − x √y m+1 √y m+1 − √x m+1