If the sum of n terms of an AP. is 4n² + 7n, then its nth term is ______.

The three geometric means between the numbers 1 and 81 are ______.

In a G.P. if the (m + n)^th terms is p and (m - n)^th terms is q then its mth terms is ______.

If the sum of first n even natural numbers is equal to m times the sum of first n is odd natural numbers then m is equal to ______.

Sum of n terms of ,the series \(\sqrt { 2 } +\sqrt { 8 } +\sqrt { 18 } +\sqrt { 32 } +\)..... is ______.

On the first day strike of physicians in a hospital, the attendance of the OPD was 1500 patients. As the strike continued, the attendance declined by 100 patients every day  . Find from which day of the strike ,the OPD would would have no patient?

Which term of the progression 19, 18 \(\frac { 1 }{ 5 } \), 17\(\frac { 2 }{ 5 } \), .....is negative term ?

Find the 12th term of a GP whose 8 th term is 192 and the common ratio is 12.

Write the first five terms of each of the sequence and obtain the corresponding series.
a_n = \({ (-1) }^{ { n }^{ 2 } }\) \(\left( \frac { { 2 }^{ { n }^{ 2 } }+3 }{ 2 } \right) \) 

If the mth term  of an AP be \({ 1 }/{ n }\) and its nth term be \({ 1 }/{ m }\) , then show that its mnth term is 1.

The sum of first three terms of a G.P. is 16 and the sum of the next three terms is 128. Determine the first term, the common ratio and the sum to n terms of the G.P.

If a,b and c be positive numbers, then prove that a² + b² + c² is greater than ab + bc + ca.

The sum of two numbers is \(\frac{13}{6} \). An even number of Am's are being inserted between them.The sum of means inserted exceeds the number of means by 1.Find the number of AM's inserted.

Between 1 and 31, m AM's have been inserted in such a way that the ratio of the 7th and (m-1)th means is 5:9,Find the value of m.

If a,b,c are in AP and b,c,d are in GP and \(\frac { 1 }{ c } ,\frac { 1 }{ d } ,\frac { 1 }{ e } \) are in AP, then prove that a,c,e are in GP.

Find the Value of n so that \(\frac { { a }^{ n+1 }+b^{ n+1 } }{ { a }^{ n }+{ b }^{ n } } \) may be the geometric mean between a and b 

The sum of two numbers is 6 times their geometric mean, show that numbers are in the ratio \(\left( 3+2\sqrt { 2 } \right) :\left( 3-2\sqrt { 2 } \right) \)

Let sum of n, 2n, 3n terms of an A.P. be S₁, S₂ and S₃ respectively, show that S₃ = 3 (S₂ - S₁)