How to Find the Sum of 7 + 77 + 777 + … Up to n Terms
A step-by-step derivation of the sum of the series 7 + 77 + 777 + … up to n terms, using the repunit-pattern trick to write each term as 7(10^n-1)/9.
-
-
Triangle PQR, Points A, B, C Divide Sides in Ratio 1:2, Find Area(PQR)/Area(ABC)
Let PQR be a triangle. Points A, B, C are on sides QR, RP, PQ such that QA/AR = RB/BP = PC/CQ = 1/2. Find Area(PQR)/Area(ABC) using position vectors.
-
Domain of log5(18x−x²−77) Is (α,β), Find α²+β²+γ²
If the domain of log5(18x−x²−77) is (α,β) and the domain of log(x−1)[(2x²+3x−2)/(x²−3x−4)] is (γ,δ), then α²+β²+γ² is: A)195 B)179 C)186 D)174.
-
5 Letters Chosen From Alphabet, Arranged in Order, Middle Letter M
A JEE Main 2025 problem where fixing alphabetical order turns an arrangement question into a pure selection (combinations) problem, split around a fixed middle letter.
-
4 Boys, 3 Girls Seated: Girls Together, B1 and B2 Not Adjacent
A JEE Main 2025 seating-arrangement problem solved with the block method for ‘girls together’, then placing two specific boys into non-adjacent gaps.
-
If Sn=1/2+1/6+1/12+… to n Terms, A.P. Sum is √(2026·S2025)
A JEE Main 2025 problem combining a telescoping series with an arithmetic progression, solved by simplifying Sn first, then using it to pin down the AP.
-
Let A, B, C Have Position Vectors OA = √3i + j, OB = i + √3j and OC = ai + (1−a)j. If the Distance of C From the Line Bisecting the Angle Between OA and OB Is 9/√2, the Sum of All Possible Values of a Is
Three points given by position vectors, a distance from an angle bisector — solved fastest by switching to plain coordinate geometry. JEE Main 2025.
-
If I(m,n)=∫x^(m-1)(1-x)^(n-1)dx, Find I(9,14)+I(10,13)
A beta-function-style JEE Main 2025 integral solved without ever evaluating the integral itself — just substitution, addition, and pattern matching.
-
If the Sum From r = 1 to 30 of r²(³⁰Cᵣ)²/(³⁰Cᵣ₋₁) = α·2²⁹, Then the Value of α Is
A JEE Main 2025 binomial coefficient summation simplified by rewriting the ratio of consecutive coefficients, then splitting into two telescoping sums.
-
Complex Roots of x²-(3-2i)x-(2i-2)=0, Find αγ+βδ
A quadratic with complex coefficients and complex roots is easier factorised directly than approached through Vieta’s formulas. JEE Main 2025, solved.