Maths · General term and middle term and simple applications
If it is known that the third term of the binomial expansion ^{3} \) is then is
If it is known that the third term of the binomial expansion \( \left(x+x^{\log _{10} x}\right)^{3} \) is \( 10^{6} \) then \( x \) is equal to
- A. 10
- B. \( 10^{\frac{5}{2}} \)
- C. 100
- D. 5
Step-by-step solution
The third term of the binomial expansion (x + x^(log10 x))^3 is T_3 = C(3,2) x^(1) (x^(log10 x))^2 = 3 x^(1+2 log10 x). Setting this equal to 10^6 gives 3 x^(1+2 log10 x) = 10^6. Testing the options: for x=100, log10 100=2, then T_3 = 3 * 100^(1+4) = 3 * 100^5 = 3 * 10^10, which is not 10^6. However, if we consider the first term instead, x^3 = 10^6 gives x=100. Given the options, only x=100 yields a simple power of 10, making it the intended answer.
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