For natural numbers `m ,n ,if(1-y)^m(1+y)^n=1+a_1y+a_2y^2+... , and a_1=a_2=10 , then`

For natural numbers m , n if (1-y)^(m)(1+y)^(n)=1+a_(1)y+a_(2)y^(2)+a_(3)y^(3)+.....,anda_(1)=a_(2)=10 , then-

For an increasing A.P. a_1, a_2, a_n ifa_1+a_3+a_5=-12 and a_1a_3a_5=80 , then which of the following is/are true? a. a_1=-10 b. a_2=-1 c. a_3=-4 d. a_5=+2

Suppose a_1, a_2, are real numbers, with a_1!=0. If a_1, a_2,a_3, are in A.P., then (a) A=[(a_1,a_2,a_3),(a_4,a_5,a_6),(a_5,a_6,a_7)] is singular (where i=sqrt(-1) ) (b)The system of equations a_1x+a_2y+a_3z=0,a_4x+a_5y+a_6z=0,a_7x+a_8y+a_9z=0 has infinite number of solutions. (c) B=[(a_1,i a_2),(ia_2,a_1)] is non-singular (d)none of these

Find the natural number a for which sum_(k=1) ^(n)f(a+k)= 16(2^(n)-1) where the function f satisfies the relation f(x+y)=f(x)f(y) for all natural numbers x,y and further f(1)=2 .

If the coefficient of-four successive termsin the expansion of (1+x)^n be a_1 , a_2 , a_3 and a_4 respectivel. Show that a_1/(a_1+a_2)+a_3/(a_3+a_4)=2 a_2/(a_2+a_3)

Let a_1, a_2 , a_3,...., a_m be the arithmetic means between -2 and 1027 and let g_1, g_2, g_3.........,g_n be the geometric mean between 1 and 1024. g_1 g_2......g_n = 2^45 and a_1 + a_2 + a_3 +....+a_m = 1025 xx 171 The value of n is :

Let a_1, a_2 , a_3,...., a_m be the arithmetic means between -2 and 1027 and let g_1, g_2, g_3.........,g_n be the geometric mean between 1 and 1024. g_1 g_2......g_n = 2^45 and a_1 + a_2 + a_3 +....+a_m = 1025 xx 171 The value of m is :

Consider the sequence defined by a_n=a n^2+b n+c dot If a_1=1,a_2=5,a n da_3=11 , then find the value of a_(10)dot

If a_1. a_2 ....... a_n are positive and (n - 1) s = a_1 + a_2 +.....+a_n then prove that (a_1 + a_2 +....+a_n)^n ge (n^2 - n)^n (s - a_1) (s - a_2)........(s - a_n)

If the coefficients of four consecutive terms in the expansion of (1+x)^n are a_1,a_2,a_3 and a_4 respectively. then prove that a_1/(a_1+a_2)+a_3/(a_3+a_4)=2a_2/(a_2+a_3).