Let f(x) be a polynomial function of second degree. Ifn f(1) = f( - 1) and a,b,c are in A.P., thenn f^'(a),f^'(b) and f^'(c) are in

(d) : Let the polynomial be f(x)=a x^2+b x+c given f(1)=f(-1) => b=0nnTherefore f(x)=a x^2+cnn nNow f^prime(x)=2 a xnnTherefore f^prime(a)=2 a^2, f^prime(b)=2 a b, f^prime(c)=2 a cnn nAs a, b, c in A . P.nn => a^2, a b, a c in A . P . => 2 a^2, 2 a b, 2 a c in A . P .nnn => f^prime(a), f^prime(b), f^prime(c) in A.P. n

Let f(x) be a polynomial function of second degree. If ... | Sequence & Series (Mathematics)

(d) : Let the polynomial be

f(x)=a x^2+b x+c

given

f(1)=f(-1) \Rightarrow b=0

\n\therefore f(x)=a x^2+c\n

\n \nNow

f^{\prime}(x)=2 a x

\n\therefore f^{\prime}(a)=2 a^2, f^{\prime}(b)=2 a b, f^{\prime}(c)=2 a c\n

\n \nAs

a, b, c \in A . P

.\n

\n\Rightarrow a^2, a b, a c \in A . P . \Rightarrow 2 a^2, 2 a b, 2 a c \in A . P .\n

\n\Rightarrow f^{\prime}(a), f^{\prime}(b), f^{\prime}(c) \in \text { A.P. }\n

Explanation

Let a_{1},a_{2},,a_{n} be a given A.P. whose common difference is an integer and S_{n} = a_{1} + a_{2} + + a_{n}. If a_{1} = 1,a_{n} = 300 and 15 \leq n \leq 50, then the ordered pair ( S_{n - 4},a_{n - 4} ) is equal to

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