Let and terms of a non-constant A. P. be respectively the and terms of a G. P. If the\\ first term of the A. P. is 1, then the sum of its first 20 terms is equal to...

aligned& 1+d, 1+7 ~d, 1+43 ~d are in GP \\& (1+7 ~d)^2=(1+d)(1+43 ~d) \\& 1+49 ~d^2+14 ~d=1+44 ~d+43 ~d^2 \\& 6 ~d^2-30 ~d=0 \\& ~d=5 \\& ~S_20=202[2 1+(20-1) 5] \\& =10[2+95] \\& =970aligned...

Let 2^{{nd}{~}},8^{{th}{~}} and 44^{{th}{~}} terms of a non-constant A. P. be respectively the 1^{{st}{~}},2^{{nd}{~}} and 3^{{rd}{~}} terms of a G. P. If the first term of the A. P. is 1, then the sum of its first 20 terms is equal to: Sequence & Series (Mathematics)

Let

2^{\text{nd}\text{~}},8^{\text{th}\text{~}}

and\n

44^{\text{th}\text{~}}

terms of a non-constant A. P. be\n respectively the

1^{\text{st}\text{~}},2^{\text{nd}\text{~}}

and\n

3^{\text{rd}\text{~}}

terms of a G. P. If the\\\n first term of the A. P. is 1, then the sum of its first 20 terms is\n equal to

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

Different A.P. are constructed with the first term 100 , the last term 199, and integral common differences. The sum of the common differences of all such A.P. having at least 3 terms and at most 33 terms is

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_{1},G_{1} and H_{1} denote the arithmetic, geometric and harmonic means . respectively of two distinct positive numbers. For n ? 2 let A_{n - 1} and H_{n - 1} have . arithmetic, geometric and harmonic means as A_{n},G_{n},H_{n} respectively. Which of the following statements is correct?

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

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

If a_1, a_2, a_n are in A.P. where a_i>0 for all i show that &\frac{1}{\sqrt{a_1}+\sqrt{a_2}}+\frac{1}{\sqrt{a_2}+\sqrt{a_3}}+\cdots+\frac{1}{\sqrt{a_{n-1}}+\sqrt{a_n}}&=\frac{n-1}{\sqrt{a_1}+\sqrt{a_n}}

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

If a_{n} = \frac{3}{4} - ( \frac{3}{4} )^{2} + ( \frac{3}{4} )^{3} + ( - 1)^{n - 1}( \frac{3}{4} )^{n} and b_{n} = 1 - a_{n}, then find the least natural number n_{0} such that b_{n} > a_{n}\forall n ? n_{0}

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