Explore sequences and series concepts for Grade 12 level math, including finding the sum of an arithmetic progression, deriving sum formulas, and practical examples. Learn how to apply these formulas to solve problems effectively.
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SEQUENCES & SERIES SEQUENCES & SERIES Gr 12 Gr 12 PART 2 LEVEL 1&2 Questions
Finding the SUM of an AP ? ?[?? + ? ? ?] ??= Sn = sum of n terms a = first term n = number of terms d = common difference Sum of an Arithmetic Sequence
Deriving the Sum formula of an AP Write out the Sum of an AP, using the general term notation (i.e. T1, T2, T3 Tn) call it Equation (1): Sn = a + (a+d) + (a+2d) + + [a+(n-2)d} + [a+(n-1)d] (1) Rewrite the Sum of n terms of an AP, but in reverse call it Equation 2: Sn = [a+(n-1)d] + [a+(n-2)d} + + (a+2d) + (a+d) + a (2)
Now add Equation 1 and Equation 2: 2 Sn = [2a+(n-1)d] + [2a+(n-1)d] + [2a+(n-1)d] + + [2a+(n-1)d] + [2a+(n-1)d] So, if there are n terms, you have n times [2a+(n-1)d] terms: 2 Sn = n [2a+(n-1)d] Solving for Sn: ? ? [?? + (? ?)?] Proof of an Arithmetic Series ?? =
Alternative SUM formula of an AP ? ?[?? + ? ? ?] ?? = ? ?? + ? + ? ? ? = ? ?[? + ? + ? ? ? ] = Sn = sum of n terms a = first term n = number of terms l = last term ? ?? + ?? = ? ?? + ? =
Deriving the Sum formula of an AP If they ask you to derive the formula, using the last term, where l = [a + (n-1)d]: Write out the Sum of an AP, using the general term notation (i.e. T1, T2, T3 Tn) call it Equation (1): Sn = a + (a+d) + (a+2d) + + (l d) + l (1) Rewrite the Sum of n terms of an AP, but in reverse call it Equation 2: Sn = l + (l d) + + (a+2d) + (a+d) + a . (2)
Now add Equation 1 and Equation 2: 2 Sn = (a+l) + (a+l) + (a+l) + + (a+l) + (a+l) So, if there are n terms, you have n times (a+l) terms: 2 Sn = n(a+l) Solving for Sn: ? ? (? + ?) ?? =
Example 1: Sibu s parents put R100 into an account on the day that he was born and then decided to continue this every year but to increase the amount by R50 annually until he turns 21. How much will he receive on his 21st birthday. 100 , 150 , 200 . S21 i.e. a = 100; n = 21 and d = 50 ? ?[?? + ? ? ?] ?? ?[?(???) + ?? ? (??)] = ??? ??? ?? = ??? =
Example 2: Determine the first term of an arithmetic progression if the sum of forty terms is 1660 and the last term is 77. i.e. a = ?; n = 40; l = 77 and S40 = 1660 ? ?[? + ?] ?? ?[? + ??] ???? = ??? + ???? ??? = ??? ? = ?? ?? = ???? =
Sigma () Notation ?(?? ?) e.g. ?=? means the SUM of iis the start of the counter nis the end of the counter (4i 3) is the general term Working with Sigma Notation So, substitute i = 1 into the general term & calculate its value; then the counter clicks up one (i.e. 2); so substitute i = 2 and find the value of the term, all the way to the nth term. Now sum up all the terms.
Example 1: Calculate: ?=? ??(?? + ?) Step 1: Find n n = top counter bottom counter + 1 = 23 3 + 1 = 21 Step 2: Find the first three terms, by substituting in the value of i a = 8(3) + 2 = 26 T2 = 8(4) + 2 = 34 T3 = 8(5) + 2 = 42
??(?? + ?) Calculate: ?=? Step 3: Find d [From Step 2: a = 26; T2 = 34 and T3 = 42] d = 8 Step 4: Substitute into the sum formula [From steps: n = 21; a = 26; d = 8] ? ?[?? + ? ? ?] ?? = ?? ?[?(??) + ?? ? (?)] = ???? ??? =
Example 2: Determine the other possible value of n, given that ?=? (?? ??) = ?=? ?=? (?? ??) ? ? (?? ??) ? a = 4(1) 20 = -16 T2 = 4(2) 20 = -12 S2 = -16 + (-12) = -28 ? ?? = ?=? (?? ??)
? ?? = ?=? (?? ??) n = ? a = 4(1) 20 = -16 T2 = 4(2) 20 = -12 T3 = 4(3) 20 = -8 d = -12 (-16) = 4 Sn = -28 ?????????? ???? ??? ?? ??????? ??? ????? ??? ?
Finding the SUM of a GP ?(?? ?) ? ? (? > ? ; ? ?) ??= Sn = sum of n terms a = first term n = number of terms r = common ratio
Deriving the Sum formula of a GP Write out the Sum of a GP, using the general term notation (i.e. T1, T2, T3 Tn) call it Equation (1): ?? = ? + ?? + ??? + + ??? ?+ ??? ? (1) Now multiply both sides of Equation 1 by r call it Equation 2: ??? = ?? + ???+ ??? + + ??? ?+ ??? (2)
Now subtract Equation 1 from Equation 2: ?? = ? + ?? + ??? + + ??? ?+ ??? ? (1) ??? = ?? + ???+ ??? + ??? ??= ? + ? + ? + + ? + ??? + ??? ?+ ??? (2) Simplify and take out a common factor: ??? ??= ??? ? ?? ? ? = ?(?? ?) Proof of a Geometric Series Solving for Sn: ?(?? ?) ? ? Sum of a Geometric Sequence ?? = (? ?)
Example 1: Determine the sum of the first ten terms, given the geometric sequence: -4; 8; -16; i.e. a = -4; n = 10; r = -2 and ???=? ?(?? ?) ? ? ? [( ?)?? ?) ( ?) ? ?? = ??? = = ???? Working with Geometric Progressions
Example 2: Determine x and then the sum of the first twelve terms, given: 1; ?; ? ? ?; ?2 ?1 =?3 ? = ?2 x ? ? =? ? ? ? ? ?= ? ?? ??= ?? ? = ? ????????: ?;? ?;? ?;
????????: ?;? ?;? ?; i.e. a = 1; n = 12; r = and ???=? ?(?? ?) ? ? ?? = ? ?)?? ? ] (? ?) ? ? [ ??? = = ?,???.. ?
Characteristics of the SUM TO INFINITY In an Arithmetic Progression, the sum to infinity will result in a DIVERGING series i.e. the sum will be infinitely big or infinitely small In a Geometric Progression, the sum to infinity will result in a CONVERGING series i.e. the sum will be get closer and closer (converge) to a particular number
Requirements to find the SUM TO INFINITY Must be a Geometric Progression (as the series will converge) The common ratio must be between -1 and 1 i.e. ? < ? < ? Sum to Infinity
Finding the SUM TO INFINITY ? ( ? < ? < ?) ? = ? ? ? = sum to infinity a = first term r = common ratio Sum to infinity example
Example 1: Determine the sum to infinity of: 1 + + + i.e. a = 1; n = ; r = and ? =? ? ? = ? ? ? ? (? = ?) = ?
Example 2: For which values of x will this sequence converge? (? ?) + ( ? ?)? + ( ? ?)? + . For a converging sequence: ? < ? < ? In this sequence: ? = ? ? ? < ? ? < ? ? < ? < ? ? < ? < ?
