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Sequences & Series-Calculations Calculations with Arithmetic Sequences: Finding the nth term of an Arithmetic Sequence ππ = ππ + (π − π)π Identifying an arithmetic sequence: Example: 2, 5, 8, 11, 14 Process: Pick 2 pairs of consecutive numbers and calculate the common difference 8–5=3 11 - 8 = 3 Solution: If the answers are the same, then it’s an arithmetic sequence with d = 3 _____________________________________________________________________________ Writing the terms of an arithmetic sequence: Examples: a) Write the first 5 terms when π1 = 15, π = −5 Solution: 15, 10, 5, 0, -5 1 1 6 3 b) Write the first 5 terms when π1 = , π = π1 = π2 = 6 1 1 + 6 3 1 1 2 3 π3 = + π4 = 1 1 5 7 3 Solution: 6 , 2 , 6 , 6 , 2 1 5 1 + 6 3 7 1 6 3 π5 = + = 1 2 + 6 6 3 2 6 6 = + = = == 5 2 + 6 6 = 7 2 6 6 = + 3 1 = 6 2 Copyright © 2011 Lynda Aguirre 5 6 7 6 9 3 6 2 = = 1 Sequences & Series-Calculations Example: Find the 21st term of 0.2, 0.6, 1.0, 1.4, 1.8, . . . 1) Is this an arithmetic sequence? Find d (pick two consecutive pairs and see if d is the same #) d = 0.6 – 0.2 = 0.4 ; d = 1.0 – 0.6 = 0.4 → it has the same d, so it’s arithmetic 2) What information do we have now? d = 0.4 π1 = 0.2 π = πππππππ 3) Use the formula for finding the nth term: πππ = π. π + (ππ − π)(π. π) πππ = π. π + (ππ)π. π πππ = π. π + π. π ππ = ππ + (π − π)π plug in the values we know simplify πππ = π __________________________________________________________________________ Example: Find the number of terms given: 3, -1, -5, -9, . . ., -65 1) Is this an arithmetic sequence? Find d (pick two consecutive pairs and see if d is the same #) d = -1 – 3 = -4 ; d = -5 – (-1) = -4 → it has the same d, so it’s arithmetic 2) What information do we have now? d = -4 π1 = 3 ππ = −65 π = πππππππ 3) Use the formula for finding the nth term: ππ = ππ + (π − π)π −ππ = π + (π − π)(−π) plug in the values we know −ππ = π − ππ + π distribute the 0.4 −ππ = π − ππ + π add like terms −ππ = π − ππ solve for n (add 7 to both sides) −ππ = −ππ simplify (divide by -4) ππ = π Copyright © 2011 Lynda Aguirre 2 Sequences & Series-Calculations Example: Find ππ and d, given: ππ = π. ππ, πππ πππ = π. ππ The problem is that the formula we have for finding the nth term requires consecutive terms, the 6th and 10th term are not consecutive. So we Formula for the nth term, given a random kth term have to alter our formula as follows: Since n=k+ (n-k), we can say that ππ = ππ + (π − π)π 2) Is this an arithmetic sequence? We were given the value for the common difference, d, so we assume it’s arithmetic 2) What information do we have now? d = unknown π6 = 1.35 π10 = 2.15 π = πππππππ Let n = 10, and k = 6 3) Use the formula for finding the nth term using the kth term: ππ = ππ + (π − π)π π. ππ = π. ππ + (ππ − π)π plug in the values we know π. ππ = π. ππ + ππ subtract 1.35 from both sides π. π = ππ simplify (divide by 4) π. π = π 4) Use the formula for finding the nth term: just found ππ = ππ + (π − π)π ,with the d value we d = 0.2 π10 = 2.15 π. ππ = ππ + (ππ − π)(π. π) π. ππ = ππ + π(π. π) π. ππ = ππ + π. π plug in the values we know simplify (subtract 1.8 from both sides) π. ππ = π Copyright © 2011 Lynda Aguirre 3 Sequences & Series-Calculations Geometric Series-Calculations http://www.ltcconline.net/greenl/courses/154/seqser/geobinom.htm Geometric Sequence: (Finding the next number in the sequence) ππ = ππ ππ−π Where r is the common ratio The sum of a Geometric Series: ππ (π − ππ ) πΊπ = π−π Copyright © 2011 Lynda Aguirre 4 Sequences & Series-Calculations Partial Sum of an Arithmetic Sequence: ππ +ππ πΊπ = π ( π π ) ; if ππ ππ ππππ€π OR πΊπ = (πππ + (π − π)π ) π Copyright © 2011 Lynda Aguirre 5