Prove that for any A.P. `a_1, a_2, a_3, ,` then determinant `|a_p+a_(p+m)+a_(p+2m)2a_p+3a_(p+m)+4a_(p+2m)a_q+a_(q+m)+a_(q+2m)2a_q+3a_(q+m)+4a_(q+2m)a_r+a_(r+m)+a_(r+2m)2a_r+3a_(r+m)+4a_(r+2m)` `4a_p+9a_(p+m)+16 a_(p+2m)4a_q+9a_(q+m)+16 a_(q+2m)4a_r+9a_(r+m)+16 a_(r+2m)|=0`

" if " a_(a) ,a_(2), a_(3)….." are in A.P, then find the value of the following determinant:" |{:(a_(p)+a_(p+m) +a_(p+2m),,2a_(p)+3a_(p+m)+4a_(p+2m),,4a_(p)+9a_(p+m)+16a_(p+2m)),(a_(p)+a_(q+m)+a_(q+2m),,2a_(q)+3a_(q+m) +4a_(q+2m),,4a_(q)+9a_(q+m)+16a_(q+2m)),(a_(r)+a_(r+m)+a_(r+2m),,2a_(r)+3a_(r+m)+4a_(r+2m),,4a_(r) +9a_(r+m)+16a_(r+2m)):}|

If a_(1),a_(2)a_(3),….,a_(15) are in A.P and a_(1)+a_(8)+a_(15)=15 , then a_(2)+a_(3)+a_(8)+a_(13)+a_(14) is equal to

Let a_(1), a_(2), a_(3), ……… be terms of an A.P. if (a_(1) + a_(2) + ...... a_(p))/(a_(1) + a_(2) + ...... a_(q)) = p^(2)/q^(2) , p != q , then a_(6)/a_(21) equals

Find the sum of first 24 terms of the A.P. a_1, a_2, a_3, , if it is know that a_1+a_5+a_(10)+a_(15)+a_(20)+a_(24)=225.

If a_(1), a_(2),………, a_(50) are in G.P, then (a_(1) - a_(3) + a_(5) - ....... + a_(49))/(a_(2) - a_(4) + a_(6) - ....... + a_(50)) =

If a_(1), a_(2), a_(3), ……. , a_(n) are in A.P. and a_(1) = 0 , then the value of (a_(3)/a_(2) + a_(4)/a_(3) + .... + a_(n)/a_(n - 1)) - a_(2)(1/a_(2) + 1/a_(3) + .... + 1/a_(n - 2)) is equal to

Let a_(1), a_(2), a_(3), a_(4) be in A.P. If a_(1) + a_(4) = 10 and a_(2)a_(3) = 24 , them the least term of them is

If the sequence a_(1),a_(2),a_(3),…,a_(n) is an A.P., then prove that a_(1)^(2)-a_(2)^(2)+a_(3)^(2)-a_(4)^(2)+…+a_(2n-1)^(2)-a_(2n)^(2)=n/(2n-1)(a_(1)^(2)-a_(2n)^(2))

If a_(1),a_(2),a_(3),…. are in A.P., then a_(p),a_(q),q_(r) are in A.P. if p,q,r are in

Let a_1,a_2,a_3…., a_49 be in A.P . Such that Sigma_(k=0)^(12) a_(4k+1)=416 and a_9+a_(43)=66 .If a_1^2+a_2^2 +…+ a_(17) = 140 m then m is equal to