If `a_(1),a_(2),a_(3),"........."a_(n)` are in HP, then the expression `a_(1)a_(2)+a_(2)a_(3)+"......"+a_(n-1)a_(n)` is equal to

If a_(1),a_(2),a_(3),".....",a_(n) are in HP, than prove that a_(1)a_(2)+a_(2)a_(3)+a_(3)a_(4)+"....."+a_(n-1)a_(n)=(n-1)a_(1)a_(n)

If a_(1),a_(2),a_(3),"........",a_(n) are in AP with a_(1)=0 , prove that (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)))=(a_(n-1))/(a_(2))+(a_(2))/(a_(n-1)) .

If a_(1),a_(2),a_(3),a_(4),a_(5) are in HP, then a_(1)a_(2)+a_(2)a_(3)+a_(3)a_(4)+a_(4)a_(5) is equal to

If a_(1),a_(2),a_(3),a_(4),,……, a_(n-1),a_(n) " are distinct non-zero real numbers such that " (a_(1)^(2) + a_(2)^(2) + a_(3)^(2) + …..+ a_(n-1)^(2))x^2 + 2 (a_(1)a_(2) + a_(2)a_(3) + a_(3)a_(4) + ……+ a_(n-1) a_(n))x + (a_(2)^(2) +a_(3)^(2) + a_(4)^(2) +......+ a_(n)^(2)) le 0 " then " a_(1), a_(2), a_(3) ,....., a_(n-1), a_(n) are in

If a_(1),a_(2),a_(3)"....." are in GP with first term a and common ratio r, then (a_(1)a_(2))/(a_(1)^(2)-a_(2)^(2))+(a_(2)a_(3))/(a_(2)^(2)-a_(3)^(2))+(a_(3)a_(4))/(a_(3)^(2)-a_(4)^(2))+"....."+(a_(n-1)a_(n))/(a_(n-1)^(2)-a_(n)^(2)) is equal to

If a_(1),a_(2),a_(3), . . .,a_(n) are non-zero real numbers such that (a_(1)^(2)+a_(2)^(2)+ . .. +a_(n-1).^(2))(a_(2)^(2)+a_(3)^(2)+ . . .+a_(n)^(2))le(a_(1)a_(2)+a_(2)a_(3)+ . . . +a_(n-1)a_(n))^(2)" then", a_(1),a_(2), . . . .a_(n) are in

If a_(1),a_(2),a_(3),…a_(n+1) are in arithmetic progression, then sum_(k=0)^(n) .^(n)C_(k.a_(k+1) is equal to (a) 2^(n)(a_(1)+a_(n+1)) (b) 2^(n-1)(a_(1)+a_(n+1)) (c) 2^(n+1)(a_(1)+a_(n+1)) (d) (a_(1)+a_(n+1))

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

STATEMENT-1 : If a, b, c are in A.P. , b, c, a are in G.P. then c, a, b are in H.P. STATEMENT-2 : If a_(1) , a_(2) a_(3) ……..a_(100) are in A.P. and a_(3) + a_(98) = 50 " then" a _(1) + a_(2) +a_(3) + ….+ a_(100) = 2500 STATEMENT-3 : If a_(1) , a_(2) , ....... ,a_(n) ne 0 then (a_(1) + a_(2)+ a_(3) + .......+ a_(n))/(n) ge (a_(1) a_(2) .........a_(n) )^(1//n)

Let a_(1)+a_(2)+a_(3), . . . ,a_(n-1),a_(n) be an A.P. Statement -1: a_(1)+a_(2)+a_(3)+ . . . +a_(n)=(n)/(2)(a_(1)+a_(n)) Statement -2 a_(k)+a_(n-k+1)=a_(1)+a_(n)" for "k=1,2,3, . . . , n