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RailwaysMathNumber SystemMedium

Which of the following fractions is less than 7/8 and greater than 1/3? निम्नलिखित में से कौन-सी भिन्न 7/8 से कम और 1/3 से अधिक है? Explanation / व्याख्या: We need: 1/3 < fraction < 7/8 Check A: 1/4 < 1/3, so incorrect. Check B: 23/24 > 7/8, so incorrect. Check C: 11/12 > 7/8, so incorrect. Check D: Compare 17/24 with 1/3: 17 × 3 = 51 24 × 1 = 24 51 > 24 Therefore, 17/24 > 1/3 Now compare 17/24 with 7/8: 17 × 8 = 136 24 × 7 = 168 136 < 168 Therefore, 17/24 < 7/8 Hence, 1/3 < 17/24 < 7/8 Answer = D Basic Arithmetic

RailwaysMathNumber SystemMedium

1/(1×4) + 1/(4×7) + 1/(7×10) + 1/(10×13) + 1/(13×16) = ? 1/(1×4) + 1/(4×7) + 1/(7×10) + 1/(10×13) + 1/(13×16) = ? Explanation / व्याख्या: We use: 1/[x(x+3)] Write: 1/[x(x+3)] = A/x + B/(x+3) Multiplying by x(x+3): 1 = A(x+3) + Bx For x = 0: 1 = 3A A = 1/3 For x = −3: 1 = −3B B = −1/3 Therefore, 1/[x(x+3)] = 1/3 [1/x − 1/(x+3)] Now apply this to every term: 1/(1×4) = 1/3(1 − 1/4) 1/(4×7) = 1/3(1/4 − 1/7) 1/(7×10) = 1/3(1/7 − 1/10) 1/(10×13) = 1/3(1/10 − 1/13) 1/(13×16) = 1/3(1/13 − 1/16) Adding: S = 1/3[(1 − 1/4) + (1/4 − 1/7) + (1/7 − 1/10) + (1/10 − 1/13) + (1/13 − 1/16)] All middle terms cancel. Therefore, S = 1/3(1 − 1/16) = 1/3 × 15/16 = 15/48 = 5/16 Hindi: सभी बीच वाले पद कट जाते हैं: S = 1/3(1 − 1/16) = 1/3 × 15/16 = 5/16

RailwaysMathNumber SystemMedium

1/2 + 1/6 + 1/12 + 1/20 + 1/30 + …… + 1/[n(n+1)] = ? 1/2 + 1/6 + 1/12 + 1/20 + 1/30 + …… + 1/[n(n+1)] = ? Explanation / व्याख्या: Observe: 1/2 = 1/(1×2) 1/6 = 1/(2×3) 1/12 = 1/(3×4) 1/20 = 1/(4×5) Therefore, the general term is: 1/[r(r+1)] Now, 1/[r(r+1)] = 1/r − 1/(r+1) Therefore, 1/(1×2) = 1 − 1/2 1/(2×3) = 1/2 − 1/3 1/(3×4) = 1/3 − 1/4 1/(4×5) = 1/4 − 1/5 ... 1/[n(n+1)] = 1/n − 1/(n+1) Hence, S = (1 − 1/2) (1/2 − 1/3) (1/3 − 1/4) ... (1/n − 1/(n+1)) All the middle terms cancel: −1/2 + 1/2 = 0 −1/3 + 1/3 = 0 −1/4 + 1/4 = 0 Therefore, S = 1 − 1/(n+1) Taking LCM: S = [(n+1) − 1]/(n+1) = n/(n+1) Hindi: प्रत्येक पद को इस प्रकार लिख सकते हैं: 1/[r(r+1)] = 1/r − 1/(r+1) अतः सभी बीच के पद कट जाते हैं और बचता है: 1 − 1/(n+1) = n/(n+1)

RailwaysMathNumber SystemMedium

Which one of the following will divide (3²⁵ + 3²⁶ + 3²⁷ + 3²⁸) completely? निम्नलिखित में से कौन (3²⁵ + 3²⁶ + 3²⁷ + 3²⁸) को पूर्णतः विभाजित करेगा? Explanation / व्याख्या: Take 3²⁵ common: 3²⁵ + 3²⁶ + 3²⁷ + 3²⁸ = 3²⁵(1 + 3 + 9 + 27) = 3²⁵ × 40 Now, 40 = 2³ × 5 And 3²⁵ contains factor 3. Therefore, the expression contains: 2 × 3 × 5 = 30 Hence it is divisible by 30.