Let a_1,a_2,a_3, be terms of an A.P. If a_1 + a_2 + + a_pa_1 + a_2 + + a_q = p^2q^2,p q, then a_6a_21 equals

(d) : Given a_1, a_2, a_3, be terms of A.P. \\&aligned& a_1+a_2++a_pa_1+a_2++a_q=p^2q^2 p2[2 a_1+(p-1) d]q2[2 a_1+(q-1) d]=p^2q^2 \\& 2 a_1+(p-1) d2 a_1+(q-1) d=pq \\& [2 a_1+(p-1) d] q=p[2 a_1+(q-1) d] \\& 2 a_1(q-p)=d[(q-1) p-(p-1) q] \\& 2 a_1(q-p)=d(q-p) 2 a_1=d \\& a_6a_21=a_1+5 da_1+20 d=a_1+10 a_1a_1+40 a_1 a_6a_21=1141aligned

Let a_{1},a_{2},a_{3}, be terms of an A.P. If \frac{a_{1} + a_{2} + + a_{p}}{a_{1} + a_{2} + + a_{q}} = \frac{p^{2}}{q^{2}},p eq q, then \frac{a_{6}}{a_{21}} equals: Sequence & Series (Mathematics)

(d) : Given a_1, a_2, a_3, \ldots \text { be terms of A.P. }\\\n&\begin{aligned}\n& \frac{a_1+a_2+\cdots+a_p}{a_1+a_2+\cdots+a_q}=\frac{p^2}{q^2} \Rightarrow \frac{\frac{p}{2}\left[2 a_1+(p-1) d\right]}{\frac{q}{2}\left[2 a_1+(q-1) d\right]}=\frac{p^2}{q^2} \\\n& \Rightarrow \frac{2 a_1+(p-1) d}{2 a_1+(q-1) d}=\frac{p}{q} \\\n& \Rightarrow\left[2 a_1+(p-1) d\right] q=p\left[2 a_1+(q-1) d\right] \\\n& \Rightarrow 2 a_1(q-p)=d[(q-1) p-(p-1) q] \\\n& \Rightarrow 2 a_1(q-p)=d(q-p) \Rightarrow 2 a_1=d \\\n& \Rightarrow \frac{a_6}{a_{21}}=\frac{a_1+5 d}{a_1+20 d}=\frac{a_1+10 a_1}{a_1+40 a_1} \Rightarrow \frac{a_6}{a_{21}}=\frac{11}{41}\n\end{aligned}

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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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

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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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