The geometric sum formula is defined as the formula to calculate the sum of all the terms in the geometric sequence. The geometric sum formula is used to calculate the sum of the terms in the geometric sequence. , ar n-1. A geometric sum is the sum of the terms in the geometric sequence. A geometric sequence with the first term a and the common ratio r and has a finite number of terms is commonly represented as a, ar, ar 2. A geometric sequence is a sequence where every term has a constant ratio to its preceding term. After how many years do Tom’s savings exceed £6000 (excluding any interest he has earned) □= □=1.1 □□= □□□( □.□ □ −□) □.□−□ >6000 Becomes □.Before going learn the geometric sum formula, let us recall what is a geometric sequence. Each year he increases the amount he put in the bank and saves by 10%. WB14 percentages exam Q Tom saves money every year. R = ¾ a = 288 So the least number of terms is n = 8ġ1 So the least number of years is n = 19 WB 12b Counter example continued □□= □(□−□□) □−□ □□=□ □ □−□ Find the sum of the following series: … + … + 1 □=□□□□, □=− □.□, □= ? 1= 1024(−0.5) □−1 1= 1024(0.5) □−1 If we work through as before, using logs, we will get the same answer for n (11) Remember at this point that you are just working out the number of terms… The negative value here does not affect how many terms there are, it just makes the pattern alternate between positive and negative numbers So if it were just ‘0.5’ rather that ‘-0.5’, we will still have the correct answer for n (the number of terms) So you can actually just use 0.5 for now, as it is only the absolute value that is important here (The negative will be important when we come to work out the sum of the sequence…) □ □ = □ 1− □ □ 1−□ □ □ = − (−0.5) −(−0.5) □ □ =683 The negative value of -0.5 DOES affect the sum of the sequence, so we do need to include it now! WB11 Using the formula □□= □(□−□□) □−□ □□=□ □ □−□ The third term of a geometric series is 5625 and the sixth term is 1215 Show that the common ratio of the sequence is 3 5 Find the first term of the sequence Find the sum of the first 18 terms of the sequence Wb9 Proof Follow these instructions: Write Sn = the first five and last algebraic terms of a geometric series Write the result of the first line with every term multiplied by the common ratio (r) Subtract the 2nd line from the first Factorise the LHS and RHS Rearrange so that LHS is Sn 1 Multiply all terms by r 2 Factorise both sides Divide by (1 - r) □ □ = □ □ □ −1 □−1Ĥ WB10 Using the formula □□= □(□−□□) □−□ □□=□ □ □−□ a) A geometric sequence has first term 20 and common ratio ¾ Find the sum of the first ten terms of the series, Giving your answer to 3 dp □ 10 = 20 1− − =75.495 □=20 □= □=10 b) A geometric sequence has first term 108 and common ratio 5 4 Find the sum of the first twelve terms of the series, Giving your answer to 3 dp □ 12 = − −1 = □= □= □=12ĥ Show that the common ratio of the sequence is 3 5 Geometric Series KUS objectives BAT work out the sum of a Geometric Sequence BAT solve problems using the formula for Sn Starter: Write the expression for the nth term of an arithmetic series Write the formula for the sum of an arithmetic series Write the instructions for the proof of the formula for the sum of an arithmetic series Presentation on theme: "Geometric Series."- Presentation transcript:Ģ Write the expression for the nth term of an arithmetic series
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