Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
KVPY2011MathematicsSequences and Series

The expression 2²+1 2²-1 + 3²+1 3²-1 + 4²+1 4²-1 + + (2011)²+1 (2011)²-1 lies in the interval

Options

  1. A(2010,2010 1 2 )
  2. B(2011- 1 2011 , 2011- 1 2012 )
  3. C(2011,2011 1 2 )
  4. D(2012,2012 1 2 )

Correct answer

C. (2011,2011 1 2 )

Step-by-step solution

2²+1 2²-1 + 3²+1 3²-1 + 4²+1 4²-1 + .+ (2011)²+1 (2011)²-1 _ r=2 ²⁰¹¹ r²+1 r²-1 = _ r=2 ²⁰¹¹ [1+ 2 (r+1)(r-1) ] array l = _ r=2 ²⁰¹¹ [1+ 1 r-1 - 1 r+1 ] =2010+ [1- 1 3 + 1 2 - 1 4 + 1 3 - 1 5 + .+ 1 2010 - 1 2012 ] =2010+1+ 1 2 - 1 2012 - 1 2011 =2011+ 1 2 - [ 1 2011 + 1 2012 ] array lies between (2011,2011 1 2 )

Practice Sequences and Series on Quantrex Academy →

More from Sequences and Series

Every term of a geometric progression is positive , and every term is the sum of the two preceding terms. Then the common ratio of the geometric progression is: 2026If k is the arithmetic mean of two given quantities and p, q are the geometric means between the same two quantities, then p^3 + q^3 is: 2026Let ' a ' and ' b ' be two numbers where a < b . The geometric mean of these numbers exceeds the smaller number by 12 and the arithmetic mean is smaller than the larger number by 2 2026The product of three numbers in geometric progression is 8 and the sum of the product of the numbers taken in pairs is 14. Find the numbers. 2026Let a₁, a₂, a₃, be a geometric progression of positive integers such that a₁ = 3 and a_ n+2 - 2a_n = a_ n+1 for all positive integers n . What is the value of a₁ + a₂ + a₃ + a₄ + a 2026The angles of a triangle are in A.P and the greatest angle is double the least angle, then sine of the third angle is 2026If we insert two numbers between 2 and 4 so that the resulting sequence is in G.P, then the inserted numbers in the order are 2026If S _ n =1^3+2^3+ . .+ n ^3 and T _ n =1+2+ . .+ n , then 2025 Full Sequences and Series list All KVPY PYQs