The first and second terms of an arithmetic progression are $$\(\frac{1}{\cos ^{2} \theta}\)$$ and $$\(-\frac{\tan ^{2} \theta}{\cos ^{2} \theta}\)$$, respectively, where $$\(0<\theta<\frac{1}{2} \pi\)$$. Find the exact value of the 13 th term when $$\(\theta=\frac{1}{6} \pi\)$$. ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................
Exam No：9709_w20_qp_13 Year：2020 Question No：7(b)
Answer:
\(a=\frac{4}{3}, d=-\frac{16}{9}\)
\(u_{13}=\frac{1}{\cos ^{2} \theta}-\frac{12}{\cos ^{4} \theta}=\frac{4}{3}+12\left(\frac{-16}{9}\right)\)
\(-20\)
\(u_{13}=\frac{1}{\cos ^{2} \theta}-\frac{12}{\cos ^{4} \theta}=\frac{4}{3}+12\left(\frac{-16}{9}\right)\)
\(-20\)
Knowledge points:
1.5.2 use the exact values of the sine, cosine and tangent of 30°, 45°, 60°, and related angles
1.6.3 use the formulae for the nth term and for the sum of the first n terms to solve problems involving arithmetic or geometric progressions (Including knowledge that numbers a,b,c are 'in arithmetic progression' if 2 b=a+c (or equivalent) and are 'in geometric progression' if (or equivalent) (Questions may involve more than one progression.)
Solution:
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