Geometric Sequences and Series.ppt
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This document presents information about geometric sequences and series. It defines geometric sequences as sequences where each term is found by multiplying the previous term by a common ratio. Geometric series are the sums of the terms in a geometric sequence. Formulas are provided for calculating individual terms, the common ratio, and the sum of finite geometric series. Applications of geometric series discussed include calculating interest, average increases, net present values, and loan repayments. Several example problems demonstrate using the formulas to find terms, common ratios, and sums.
Geometric Sequences and Series.ppt
- 1. Presentation Title: Applications of Geometric series in Business. Prepared By: Wasif Ali Syed (18221554-001)
- 2. Sequence A Sequence is a set of things (usually numbers) that are in order.
- 3. Series The sum of an indicated number of terms in a sequence is called a series. e.g., the sum of the first seven terms of the sequence is the series. 4+8+12+16+20+24+28
- 4. Geometric Sequences A sequence {an}n=0 = {a0, a1 ,a2 ,a3 , …}is an ordered set of numbers. The index of each term of the sequence indicates the position or order in which specific data is found. This order is very important. For example, the sequence {1,3,5,7,9, …} differs from the sequence {9,7,5,3,1, …} , even if the terms are the same.
- 5. Geometric series Given {an} = {a0, a1r ,a2r2,a3r3, …}, a geometric sequence of common ratio r. A geometric series is the sum of the elements of a geometric sequence a0 + a1r + a2r2 + a3r3 +…. A series can be finite (with a finite number of terms) or infinite.
- 6. Formula to evaluate n term in geometric sequence an = a1rn – 1 Where a1 is the first term r is the common ratio n is the number of terms
- 7. Formula to evaluate common ratio in geometric sequence Where r is the common ratio a2 is the 2nd term. a1 is the 1st term. r a a 2 1
- 8. Formula to evaluate a geometric series Where Sn Sum of GP with n term a1 first term n number of terms r common ratio
- 9. Application of Geometric Series 1. It is used in the calculation of interest. 2. It is used to find the average increase in sale, production, or other economic or business data. 3. It is theoretically considered to be the best average in the construction of index number.
- 10. Application of Geometric Series 4. It is used in accountancy in finding the Net Present Value of projects. 5. It is used in calculation of repayments of loans and values of investments.
- 11. Question #1 Write the first five terms of the geometric sequence whose first term is a1 = 9 and r = (1/3). Solution an = a1rn – 1 a2 = (9)(1/3)2 – 1 = 3 an = 9,3,1,1/3,1/9 a3 = (9)(1/3)3 – 1 = 1 a4 = (9)(1/3)4 – 1 = 1/3 a5 = (9)(1/3)5 – 1 = 1/9
- 12. Question #2 Find the 15th term of the geometric sequence whose first term is 20 and whose common ratio is 1.05 Solution an = a1rn – 1 a15 = (20)(1.05)15 – 1 a15 = 39.599
- 13. Question #3 Find a formula for the nth term 5, 15, 45, … Solution What is the 9th term? an = a1rn – 1 an = 5(3)n – 1 an = 5(3)n – 1 a9 = 5(3)8 a9 = 32805
- 14. Question #4 Find the sum of the first 12 terms of the series 4(0.3)n = 4(0.3)1 + 4(0.3)2 + 4(0.3)3 + … + 4(0.3)12 3 . 0 1 ) 12 3 . 0 ( 2 . 1 2 . 1 12 S = 1.714 r r a a S n n 1 1 1
- 15. Question #5 Find the sum of the first 5 terms of the series 5/3 + 5 + 15 + … r = 5/(5/3) = 3 r r a a S n n 1 1 1 3 1 ) 5 3 ( 3 5 3 5 5 S = 605/3
- 16. Thank you