Let 6 Arithmetic means A1, A2, A3, A4, A5, As are inserted between two consecutive natural number and b (a > b). If A12 - A22 + A32 - A1+ As2 - As? is equal to prime number then 'b' is equal to(A) 1(B) 2C3(D) 4​

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RiaElidaMF RiaElidaMF Tuteehub forum best answer Best Answer 1 year ago

rithmetic means A1, A2, A3, A4, A5, As are inserted between two consecutive natural number and b (a > b). If A12 - A22 + A32 - A1+ As2 - As? is EQUAL to prime number then 'b' is equal toGiven,6 Arithmetic means A1, A2, A3, A4, A5, A6If N AMs are inserted between two numbers a and b, then we have common difference,d = ( b - a ) / (N + 1)as N = 6, we have,d = ( b - a ) / (6 + 1)d = ( b - a ) / 7Now consider,A1² - A2² + A3² - A4²+ A5² - A6²= (A1 + A2) (A1 - A2) + (A3 + A4) (A3 - A4) + (A5 + A6) (A5 - A6) = (A1 + A2) (-d) + (A3 + A4)  (-d)  + (A5 + A6)  (-d) = (-d) (A1 + A2 + A3 + A4 + A5 + A6)as we know that,A1 = a + dA2 = a + 2dA3 = a + 3dA4 = a + 4dA5 = a + 5dA6 = a + 6d= (-d) (a + d + a + 2d + a + 3d + a + 4d + a + 5d + a + 6d)= (-d) (6a + 21d)= - ( b - a ) / 7 [ 6a + 21 ( b - a ) / 7 ]= (a - b)/7 (21a + 21b)/7= (a - b)/7  × 3(a + b)= 3/7 (a - b) (a + b)= 3/7 (a² - b²)Given 3/7 (a² - b²) is a prime number This is POSSIBLE only when,  (a² - b²) = 7 Given a and b are consecutive numbers and a > b. Therefore, a = b + 1using the above equation, we GET, (a² - b²) = 7 (b+1)² - b² = 7b² + 1 + 2b  - b² = 71 + 2b = 72b = 6b = 3Therefore the values of a and b are a = b + 1 = 3 + 1 = 4 and b = 3∴ a = 4, b = 3


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