If 3^2sin2 - 1,14 and 3^4 - 2sin2 are the first three terms of an A.P. for some , then the sixth term of this A.P. is

(a) I Since, the given terms are first three terms of an A.P \\ 2 11=3^2 2 x+3^4-2 2 x \\ Putting, 3^2 sing =x, we get 28=x3+81x \\ 28=x^2+2433 x \\ &n

If 3 (2sin2 alpha 1) 14 and 3 (4 2sin2 alpha) are the first three terms of an A.P. for some alpha then the sixth term of this A.P. is: 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

Let a_{1},a_{2},a_{3},. be a G.P. of increasing positive numbers. Let the sum of its 6^{{th}{~}} and 8^{{th}{~}} terms be 2 and the product of its 3^{{rd}{~}} and 5^{{th}{~}} terms be 1/9. Then 6( a_{2} + a_{4} )( a_{4} + a_{6} ) is equal to

For any odd integer n ? 1,n^{3} - (n - 1)^{3} + + ( - 1)^{n - 1}1^{3} =

The sum of the first n terms of the series (1^{2} + 2.2^{2} + 3^{2} + 2.4^{2} + 5^{2} + 2 . 6^{2} + ..) is (\frac{n(n + 1)^{2}}{2}, when n is even. when n is odd, the sum is

Let a,b be two non-zero real numbers. If p and r are the roots of the equation x^{2} - 8ax + 2a = 0 and q and s are the roots of the equation \end{enumerate} x^{2} + 12bx + 6b = 0, such that \frac{1}{p},\frac{1}{q},\frac{1}{r},\frac{1}{s} are in A.P., then a^{- 1} - b^{- 1} is equal to

If a_{1},a_{2},a_{3}, are in A.P. such that a_{1} + a_{7} + a_{16} = 40, then the sum of the first 15 terms of this A.P. is

Let \frac{1}{x_{1}},\frac{1}{x_{2}},,\frac{1}{x_{n}}( x_{i} eq 0 .\ for . \ i = 1,2,,n ) be in A.P. such that x_{1} = 4 and x_{21} = 20. If n is the least positive integer for which x_{n} > 50, then \sum_{i = 1}^{n}\mspace{2mu}( \frac{1}{x_{i}} ) is equal to

Let 3,a,b,c be in A.P. and 3,a - 1,b + 1,c + 9 be in G.P. Then, the arithmetic mean of a,b and c is

Number of 4 -digit numbers that are less than or equal to 2800 and either divisible by 3 or by 11 , is equal to

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