Question:
What is the value of \(\Delta^n e^{ax+b}\)?
\(\Delta^n e^{ax+b}\) का मान क्या है?
Options:
- \((e^{ah} – 1)^n e^{ax – b}\)
- \((e^{ah} – 1)^n e^{ax + b}\)
- \((e^{ah} + 1)^n e^{ax – b}\)
- \((e^{ah} + 1)^n e^{ax + b}\)
- \((e^{ah} – 1)^n e^{ax – b}\)
- \((e^{ah} – 1)^n e^{ax + b}\)
- \((e^{ah} + 1)^n e^{ax – b}\)
- \((e^{ah} + 1)^n e^{ax + b}\)
Correct Answer: b)
Explanation:
Using the forward difference operator \(\Delta = E – 1\) with interval of differencing \(h\):
\[ \Delta e^{ax+b} = e^{a(x+h)+b} – e^{ax+b} = e^{ax+b}(e^{ah} – 1) \]
Applying \(\Delta\) repeatedly \(n\) times:
\[ \Delta^n e^{ax+b} = (e^{ah} – 1)^n e^{ax+b} \]
अग्र अंतर ऑपरेटर (forward difference operator) \(\Delta = E – 1\) के अनुसार:
\[ \Delta e^{ax+b} = e^{a(x+h)+b} – e^{ax+b} = e^{ax+b}(e^{ah} – 1) \]
इसे \(n\) बार लागू करने पर:
\[ \Delta^n e^{ax+b} = (e^{ah} – 1)^n e^{ax+b} \]