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blondinia [14]
3 years ago
10

The sum of the first three terms of an arithmetic sequence is 108, and the sum of the next three terms is 183. what is the value

of the 11th term
Mathematics
1 answer:
qaws [65]3 years ago
4 0
The key is to find the first term a(1) and the difference d.

in an arithmetic sequence, the nth term is the first term +(n-1)d
the firs three terms: a(1), a(1)+d, a(1)+2d
the next three terms: a(1)+3d, a(1)+4d, a(1)+5d,
a(1) + a(1)+d +a(1)+2d=108
a(1)+3d + a(1)+4d + a(1)+5d=183

subtract the first equation from the second equation: 9d=75, d=75/9=25/3
Plug d=25/3 in the first equation to find a(1): a(1)=83/3

the 11th term is: a(1)+(25/3)(11-1)=83/3 +250/3=111

Please double check my calculation. <span />
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If f(x) = 9x10 tan−1x, find f '(x).
djverab [1.8K]

Answer:

\displaystyle f'(x) = 90x^9 \tan^{-1}(x) + \frac{9x^{10}}{x^2 + 1}

General Formulas and Concepts:

<u>Calculus</u>

Differentiation

  • Derivatives
  • Derivative Notation

Derivative Property [Multiplied Constant]:                                                           \displaystyle \frac{d}{dx} [cf(x)] = c \cdot f'(x)  

Basic Power Rule:

  1. f(x) = cxⁿ
  2. f’(x) = c·nxⁿ⁻¹

Derivative Rule [Product Rule]:                                                                             \displaystyle \frac{d}{dx} [f(x)g(x)]=f'(x)g(x) + g'(x)f(x)

Step-by-step explanation:

<u>Step 1: Define</u>

<em>Identify</em>

\displaystyle f(x) = 9x^{10} \tan^{-1}(x)

<u>Step 2: Differentiate</u>

  1. [Function] Derivative Rule [Product Rule]:                                                   \displaystyle f'(x) = \frac{d}{dx}[9x^{10}] \tan^{-1}(x) + 9x^{10} \frac{d}{dx}[\tan^{-1}(x)]
  2. Rewrite [Derivative Property - Multiplied Constant]:                                  \displaystyle f'(x) = 9 \frac{d}{dx}[x^{10}] \tan^{-1}(x) + 9x^{10} \frac{d}{dx}[\tan^{-1}(x)]
  3. Basic Power Rule:                                                                                         \displaystyle f'(x) = 90x^9 \tan^{-1}(x) + 9x^{10} \frac{d}{dx}[\tan^{-1}(x)]
  4. Arctrig Derivative:                                                                                         \displaystyle f'(x) = 90x^9 \tan^{-1}(x) + \frac{9x^{10}}{x^2 + 1}

Topic: AP Calculus AB/BC (Calculus I/I + II)

Unit: Differentiation

7 0
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Last one, So so sorry!
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