- Level: International Baccalaureate
- Subject: Maths
- Word count: 1531
MAth portfolio-Infinite surds
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Introduction
Q1) Find a formula for an+1 in terms of an. Answer) The following expression is an example of an infinite surd: This means that can me shown as= a1 = a2 = a3 = A formula for an+1 in terms of an would be: a2 = a3 = Square on both sides (a3)2 = ()2 (a3)2 = 1 + a2 a2 = (a3)2 - 1 an = (an+1)2 - 1 (an+1)2 = an + 1 Therefore, an+1 = Q2) Calculate the decimal values of the first ten terms of the sequence. Using technology, plot the relation between 'n' and 'a'. Describe what you notice. What does this suggest about the value of an - an+1 as 'n' gets very large? Use your results to find the exact value for this infinite surd. Ans) > n Term (an) an - an+1 > 1 1.414213562 - > 2 1.553773974 0.139560412 > 3 1.598053182 0.044279208 > 4 1.611847754 0.013794572 > 5 1.616121207 0.004273452 > 6 1.617442799 0.001321592 > 7 1.617851291 0.000408492 > 8 1.617977531 0.000126241 > 9 1.618016542 3.90113 x 10-5 > 10 1.618028597 1.20552 x 10-5 The difference between each successive term is beginning to decrease and is almost at zero. ...read more.
Middle
Find some value of 'k' that makes the expression an integer. Find the general statement that represents all the values of 'k' for which the expression is an integer. ==> I used Apple iWork'09 Numbers to find the values of k that are integers. k a1 a2 a3 A4 A5 A6 A7 A8 A9 a10 a11 1 1.414214 1.553774 1.598053 1.611848 1.616121 1.617443 1.617851 1.617978 1.618017 1.618029 1.618032 2 1.847759 1.961571 1.990369 1.997591 1.999398 1.999849 1.999962 1.999991 1.999998 1.999999 2 3 2.175328 2.274935 2.296723 2.301461 2.30249 2.302714 2.302762 2.302773 2.302775 2.302775 2.302776 4 2.44949 2.539585 2.557261 2.560715 2.561389 2.561521 2.561547 2.561552 2.561553 2.561553 2.561553 5 2.689994 2.773084 2.788025 2.790703 2.791183 2.791269 2.791284 2.791287 2.791288 2.791288 2.791288 6 2.906801 2.984426 2.997403 2.999567 2.999928 2.999988 2.999998 3 3 3 3 7 3.105761 3.178956 3.190448 3.192248 3.19253 3.192574 3.192581 3.192582 3.192582 3.192582 3.192582 8 3.290658 3.360157 3.370483 3.372015 3.372242 3.372275 3.37228 3.372281 3.372281 3.372281 3.372281 9 3.464102 3.530453 3.539838 3.541163 3.541351 3.541377 3.541381 3.541381 3.541381 3.541381 3.541381 10 3.627985 3.69161 3.700218 3.70138 3.701538 3.701559 3.701562 3.701562 3.701562 3.701562 3.701562 11 3.783732 3.844962 3.852916 3.853948 3.854082 3.854099 3.854102 3.854102 3.854102 3.854102 3.854102 12 3.932442 3.991546 3.998943 3.999868 3.999983 3.999998 4 4 4 4 4 13 4.074991 4.13219 4.139105 4.13994 4.140041 4.140053 ...read more.
Conclusion
Based on all this, I got to my general statement, and it is; an = 2 (a(n-1) + 1) - an-2 Q6) Test the validity of your statement by using different values of 'k'. Answer) a5 = 2 (a(5-1) + 1) - a(5-2) = 2 (a4 + 1) - a3 = 2 (20 + 1) - 12 = 42 - 12 = 30 a4 = 2 (a(4-1) + 1) - a(4-2) = 2 (a3 + 1) - a2 = 2 (12 + 1) - 6 = 26 - 6 = 20 So, the general statement that we concluded is completely valid. Q7) Discuss the scope and/or limitation of your general statement. Answer) To know any of these terms, we need to know about its 2 preceding terms, and because of this, we may not be able to find the first 2 terms. They have to be given, as there are no terms that precede the first 2 terms. I only found values up to the 5th term; therefore I drew this conclusion from my analysis of the behaviour of those terms. Q8) Explain how you arrived at your general statement. Answer) As is said above, the successive term can be obtained by the relation; an = 2 (an-1 + 1) - an-2 Samonvye Reddy| {Math Portfolio} Samonvye Reddy| {Math Portfolio} 1 1 ...read more.
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