Arithmetic Series: Finding the Sum of an Arithmetic Sequence, Slides of Pre-Calculus

Download Arithmetic Series: Finding the Sum of an Arithmetic Sequence and more Slides Pre-Calculus in PDF only on Docsity! Arithmetic Series Return to Table of Contents WARMUP: Add up all the terms of the arithmetic sequence without a calculator! 1, 2, 3, 4,.....97, 98, 99, 100 Is there a shortcut??? 19 Consider 3, 9, 15, 21, 27, 33, 39,... what is S,? 20 Consider 3, 9, 15, 21, 27, 33, 39,... whatis S,? Arithmetic Series Suppose we wanted to find the first 100 terms of   4, 7,10, 13, 16, 19, 22, . . . or S100 ? There must be a short cut! The 100th  term of the sequence is 301 using a100  = 4 + (100 ­ 1)3. S100  = 4 + 7 + 10 +13 + . . . + 292 + 295 +298 + 301 If we add the smallest and largest (4 + 301)= 305 If we add the next two (7 + 298) = 305 and continue (10 + 295) until they are all paired up (151+154).   We now have 50 pairs of 305, so... S100  = 4 + 7 + 10 +. . .+298 + 301 = 50(305) = 15250 Do you understand the pattern? Find S, for each arithmetic series. eae rr errr—~—r—e—r——svrvrvrvyvyvoes Example: a,, = 30 and d = -7 Example: a, = 2, a, = 32, andd=5 21 Find S, for the arithmetic series described: a,= 19 anda, = 37. 22 Find S, for the arithmetic series described: a, = 30 anda, =-45. 25 Find S, for the arithmetic series described: 17+20+23+26+29+...+50 Sigma Notation Sigma ( >" ) is the Greek letter S. Sigma represents the sum of specific terms in a sequence. The difference between S,, and sigma is that S, is always the sum of the first to the last term. However, Sigma can begin and end with any two terms. 9 4x means start with the 3 term and sum up through the th = 9th term. x Ax = 4(3) + 4(4)+ 465) + 4(6) + 4(7) + 4(8) + 409) = 168 x=3 4 y3k = 3()+ 3(2)+ 33) + 3(4) + 3(5) = 34+.64+9412415 =45 8 7 = 2(5)+ 2(6)+ 2(7) + 2(8) = 10412414416 =52 a5 yav+1- (4(4) £1) + (4(5) +1) +(4(6) +1) = 17+ 214 25 = 63 wa4 x 2f4+8=(2(8)+8)4+C2(4)+8)+ @26)4+8)4+ 26) +8)4+ -2(7) +8) ” =2+0+-2+-4+-6 =-10 8 28 Evaluate 5° (10 1) t=5 30 How would evaluate > (4d+2) ? dae 30 >) (4d +2) =42 +46 +50+...4122 d=10 S02 +122) =10.5(164) = 246 NOTE: There are 21 terms from the 10th to the 30th. Don't be fooled and say there are only 20 terms. There are always (high - low + 1) terms. 100 29 Evaluate > (200 — 37) t=50