4.9 Geometric sequences and series (A-level only) Flashcards","topicId":"850053d0-2d92-4c05-a5ec-0f48a07d0e19","topicSlug":"geometric-sequences-and-series","cards":[{"id":"3bdd82a1-c00a-4f42-b4aa-df97254c3e35","cardType":"basic","front":"How do you test whether a sequence is geometric?","back":"Check that consecutive non-zero terms have a constant ratio.","points":1,"mcqOptions":null},{"id":"43611259-ebaa-4ee4-b5e4-7de68261f199","cardType":"basic","front":"Find the common ratio of $80, 40, 20, 10,\\ldots$","back":"$r=\\frac{40}{80}=\\frac12$","points":1,"mcqOptions":null},{"id":"19ddd5be-9d07-4457-a7cd-c362c275c7fc","cardType":"basic","front":"What does $r<0$ imply about a geometric sequence?","back":"The terms alternate in sign.","points":1,"mcqOptions":null},{"id":"3ce65bdb-6205-4a93-9981-6fa4856eb5ff","cardType":"basic","front":"What happens after the first term when $r=0$?","back":"Every subsequent term is zero.","points":1,"mcqOptions":null},{"id":"126a967a-3b06-49d9-b26d-b5fe14812ed3","cardType":"cloze","front":"For first term $a$ and common ratio $r$, the nth term is $\\text{[...]}$.","back":"For first term $a$ and common ratio $r$, the nth term is $\\boxed{u_n=ar^{n-1}}$.","points":null,"mcqOptions":null},{"id":"f10bee5c-cac3-4e55-b827-6e65a067fb75","cardType":"basic","front":"Why is the exponent $n-1$ in $u_n=ar^{n-1}$?","back":"Reaching term $n$ requires $n-1$ multiplications by $r$.","points":2,"mcqOptions":null},{"id":"6f1d9435-fa40-4ee8-a0f6-0f3e565c43f9","cardType":"basic","front":"Find the eighth term of $5,15,45,\\ldots$","back":"$u_8=5\\times3^7=10935$","points":2,"mcqOptions":null},{"id":"35a49b51-73de-4e3e-9b1f-341b0e802498","cardType":"basic","front":"Which term of $2,6,18,\\ldots$ is $1458$?","back":"The seventh term, since $2\\times3^6=1458$.","points":2,"mcqOptions":null},{"id":"c28e2f8d-87af-4ff5-9809-9e86151f331d","cardType":"cloze","front":"For $r\\ne1$, the finite geometric sum is $\\text{[...]}$.","back":"For $r\\ne1$, the finite geometric sum is $\\boxed{S_n=\\frac{a(1-r^n)}{1-r}}$.","points":null,"mcqOptions":null},{"id":"c09f6661-9620-4a65-a89a-b83c976ac5e1","cardType":"basic","front":"What is $S_n$ when a geometric series has $r=1$?","back":"$S_n=na$","points":1,"mcqOptions":null},{"id":"6916745a-663b-4e72-95b6-fa6241ff1297","cardType":"basic","front":"Find the sum of the first six terms of $7,14,28,\\ldots$","back":"$S_6=\\frac{7(2^6-1)}{2-1}=441$","points":2,"mcqOptions":null},{"id":"8850d12e-d98f-435b-a1f8-1e582957023c","cardType":"cloze","front":"An increase of $p$% gives the geometric multiplier $\\text{[...]}$.","back":"An increase of $p$% gives the geometric multiplier $\\boxed{r=1+\\frac{p}{100}}$.","points":null,"mcqOptions":null},{"id":"9c3e88de-e615-486d-83e8-1ec404ad484a","cardType":"cloze","front":"A decrease of $p$% gives the geometric multiplier $\\text{[...]}$.","back":"A decrease of $p$% gives the geometric multiplier $\\boxed{r=1-\\frac{p}{100}}$.","points":null,"mcqOptions":null},{"id":"e8071d20-258b-4e87-b40a-3f2b10b8aa04","cardType":"basic","front":"A machine initially worth £24,000 depreciates by 15% annually. What is its value after four years?","back":"$24000(0.85)^4=£12528.15$","points":2,"mcqOptions":null},{"id":"bbf74dc5-0857-4477-96a8-54d8dbed910d","cardType":"basic","front":"When does $u_n=ar^{n-1}$ converge to zero?","back":"When $|r|<1$.","points":1,"mcqOptions":null},{"id":"70c9db45-bf9d-4254-be74-3e2ce5ad65b9","cardType":"basic","front":"What is the limit of $u_n=ar^{n-1}$ when $r=1$?","back":"$a$","points":1,"mcqOptions":null},{"id":"b36b8735-4534-42d1-8960-b6954d7d8c73","cardType":"basic","front":"Why does a geometric sequence with $r=-1$ and $a\\ne0$ diverge?","back":"It alternates between $a$ and $-a$

1.4.9 Geometric sequences and series (A-level only) Flashcards

1.4.9 Geometric sequences and series (A-level only) Flashcards