Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
AP EAMCET202021 Sep 2020Morning ShiftMathematicsSequences and SeriesActual

Let (P(n): 1^2+2^2+3^2+ +n^2 ) (= 6(n-1)(n-2) (n-2020)+2 n^3+3 n^2+n 6 ), for all (n N ). Then which of the following is correct?

Options

  1. A(P(n) ) is true for all (n N )
  2. B(P(n) ) is true for all (h>2020 )
  3. C(P(n) ) is true for all (n 2020 )
  4. D(P(n) ) is not true for any (n N )

Correct answer

C. (P(n) ) is true for all (n 2020 )

Step-by-step solution

Given statement ( aligned & P(n)=1^2+2^2+3^2+ +n^2= & 6(n-1)(n-2) (n-2020)+2 n^3+3 n^2+n 6 aligned ) ( ) We know that, ( aligned & 1^2+2^2+3^2+ +n^2= n(n+1)(2 n+1) 6 & 6(n-1)(n-2) (n-2020)+2 n^3+3 n^2+n 6 & =(n-1)(n-2) (n-2020)+ n(n+1)(2 n+1) 6 aligned ) will be ( n(n+1)(2 n+1) 6 ) if (n=1,2,3, ., 2020 ) only. Therefore, (P(n) ) is true for all (n 2020 ). Hence, option (c) is correct.

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 AP EAMCET PYQs