If 3^2sin2alpha - 1,14 and 3^4 - 2sin2alpha are the first three terms of an A.P. for some alpha, then the sixth term of this A.P. is

(a) I Since, the given terms are first three terms of an A.P \ Therefore 2 x 11=3^2 min 2 x+3^4-2 min 2 x Putting, 3^2 sing =x, we get 28=(x / 3)+(81 / x) \ => 28=(x^2+243 / 3 x) \ => x^2-84 x+243=0 \ => x^2-81 x-3 x+243=0 \ => x=3,81 \ Therefore 3^2 sin 2 alpha=3 or 3^4 \ => 2 sin 2 alpha=1 or 4 \ => sin 2 alpha=(1 / 2) or 2 (rejected) First term, a=3^2 sin 2 alpha^-1=3^(1 / 2)-1=1 Second term = 14 ∴ Common difference, d=13 Now, 6^ th term, t_6=a+5 d=1+5(13)=66

If 3 (2sin2 alpha 1) 14 and 3 (4 2sin2 alpha) are the f... | Sequence & Series (Mathematics)

(a) I Since, the given terms are first three terms of an A.P $\\ \therefore 2 \times 11=3^{2 \min 2 x}+3^4-2 \min 2 x \\$ Putting, $3^2$ sing $=x$ , we get $28=\frac{x}{3}+\frac{81}{x} \\ \Rightarrow 28=\frac{x^2+243}{3 x} \\ \Rightarrow x^2-84 x+243=0 \\ \Rightarrow x^2-81 x-3 x+243=0 \\ \Rightarrow x=3,81 \\ \therefore 3^{2 \sin 2 \alpha}=3 \text { or } 3^4 \\ \Rightarrow 2 \sin 2 \alpha=1 \text { or } 4 \\ \Rightarrow \sin 2 \alpha=\frac{1}{2} \text { or } 2 \text { (rejected) } \\$ First term, $a=3^{2 \sin 2} \alpha^{-1}=3^{\frac{1}{2}-1}=1\\$ Second term = 14 ∴ Common difference, $d=13\\$ Now, $6^{\text {th }}$ term, $t_6=a+5 d=1+5(13)=66$

Explanation

Let a_{1},a_{2},,a_{n} be a given A.P. whose common difference is an integer and S_{n} = a_{1} + a_{2} + + a_{n}. If a_{1} = 1,a_{n} = 300 and 15 \leq n \leq 50, then the ordered pair ( S_{n - 4},a_{n - 4} ) is equal to

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