Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
JEE Main 202622 JanuaryEvening Shift
Exam
JEE Main
Year
2026
Date
22 January
Shift
Evening Shift
Subject
Mathematics
Chapter
Binomial Theorem

JEE Main 2026 · 22 January · Evening Shift · Mathematics · Binomial Theorem

Let C_ r denote the coefficient of x^ r in the binomial expansion...

Let Cr\mathrm{C}_{\mathrm{r}} denote the coefficient of xrx^{\mathrm{r}} in the binomial expansion of (1+x)n,n∈N,0≤r≤n(1+x)^{\mathrm{n}}, \mathrm{n} \in \mathrm{N}, 0 \leq \mathrm{r} \leq \mathrm{n}. If Pn=C0−C1+223C2−234C3+…..+(−2)nn+1CnP_{n}=C_{0}-C_{1}+\frac{2^{2}}{3} C_{2}-\frac{2^{3}}{4} C_{3}+\ldots. .+\frac{(-2)^{n}}{n+1} C_{n}, then the value of ∑n=1251P2n\sum_{n=1}^{25} \frac{1}{P_{2 n}} equals.

Options

  1. A650
  2. B675
  3. C525
  4. D580

Correct answer

B. 675

Step-by-step solution

Using the identity 1r+1(nr)=1n+1(n+1r+1)\frac{1}{r+1}\binom{n}{r} = \frac{1}{n+1}\binom{n+1}{r+1},
we get Pn=∑r=0n(−2)rr+1Cr=1n+1∑s=1n+1(−2)s(n+1s)P_n = \sum_{r=0}^{n}\frac{(-2)^r}{r+1}C_r = \frac{1}{n+1}\sum_{s=1}^{n+1}(-2)^{s}\binom{n+1}{s}.
Since (1−2)n+1=(−1)n+1=∑s=0n+1(−2)s(n+1s)(1-2)^{n+1} = (-1)^{n+1} = \sum_{s=0}^{n+1}(-2)^s\binom{n+1}{s},
we have ∑s=1n+1(−2)s(n+1s)=(−1)n+1−1\sum_{s=1}^{n+1}(-2)^s\binom{n+1}{s} = (-1)^{n+1} - 1.
For even n=2kn = 2k: P2k=1−(−1)2(2k+1)=12k+1P_{2k} = \frac{1 - (-1)}{2(2k+1)} = \frac{1}{2k+1}.
Thus 1P2n=2n+1\frac{1}{P_{2n}} = 2n+1, and ∑n=125(2n+1)=2⋅25⋅262+25=675\sum_{n=1}^{25}(2n+1) = 2 \cdot \frac{25 \cdot 26}{2} + 25 = 675.
Practice Binomial Theorem on Quantrex Academy →

More from Binomial Theorem

If 26((2^3)/(3) 12 2 + (2^5)/(5) 12 4 + (2^7)/(7) 12 6 + … + 2^ 13 13 12 12 ) = 3^ 13 - α… JEE Main 2026If (1 - x^3)^ 10 = ∑ _ r=0 ^ 10 a_r x^r (1-x)^ 30-2r , then 9a_9 a_ 10 is equal to… JEE Main 2026If the coefficients of the middle terms in the binomial expansions of (1 + α x)^ 26 and… JEE Main 2026The coefficient of x^2 in the expansion of (2x^2 + (1)/(x))^ 10 , x ≠ 0 , is : JEE Main 2026If the sum of the coefficients of x^7 and x^ 14 in the expansion of ((1)/(x^3) - x^4)^n… JEE Main 2026In the expansion of (9x- 1 3 x )^ 18 , x>0 , if the term independent of x is (221)k… JEE Main 2026Let the smallest value of k ∈ ℕ , for which the coefficient of x^3 in (1+x)^3 + (1+x)^4 +… JEE Main 2026If for 3 ≤ r ≤ 30 , 30 30-r + 3 30 31-r + 3 30 32-r + 30 33-r = m r , then m equals: JEE Main 2026The number of elements in the set S = \ (r, k) : k ∈ ℤ and ^ 36 C_ r+1 = 6( ^ 35 C_r)… JEE Main 2026Given below are two statements : Statement I: 25^ 13 +20^ 13 +8^ 13 +3^ 13 is divisible… JEE Main 2026The sum of the coefficients of x^ 499 and x^ 500 in (1+x)^ 1000 +x(1+x)^ 999 +x^ 2 (1+x)^… JEE Main 2026Let S=(1)/(25!)+(1)/(3!23!)+(1)/(5!21!)+… up to 13 terms. If 13 ~S= 2^ k n! , k ∈ ~N… JEE Main 2026The value of ^ 100 C_ 50 51 + ^ 100 C_ 51 52 +….+ ^ 100 C_ 100 101 is : JEE Main 2026The sum of all possible values of n ∈ N , so that the coefficients of x, x^ 2 and x^ 3 in… JEE Main 2026The coefficient of x^ 48 in (1+x)+2(1+x)^ 2 +3(1+x)^ 3 +…+100(1+x)^ 100 is equal to JEE Main 2026If ( 1 ^ 15 C_ 0 + 1 ^ 15 C_ 1 )( 1 ^ 15 C_ 1 + 1 ^ 15 C_ 2 ) ⋯( 1 ^ 15 C_ 12 + 1 ^ 15 C_… JEE Main 2026 Full Binomial Theorem list Binomial Theorem JEE Maths PYQsJEE Mathematics PYQs All JEE Main PYQs