Describe the similarities and differences between arithmetic and geometric sequences.

1 answer:

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Arithmetic sequence is a sequence with the difference between two consecutive terms constant. The difference is known as the common difference.

Arthmetic Sequence: 2,4,6,8,10,12. The common difference is 2. A pattern of numbers that increase or decrease at a constant amount.

A geometric sequence is a sequence with the ratio between two consecutive terms constant. This ratio is called common ratio.

Geometric Sequence: 2,10,50, 250. 2x5=10 10x5=50 and so on. R=5

A geometric sequence is one where to get from one term to the next you multiply by the same number each time. This number is called the common ration, R.

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(a) The function k is defined by k(x)=f(x)g(x). Find k′(0).

Brut [27]

Answer:

(a) k'(0) = f'(0)g(0) + f(0)g'(0)

(b) m'(5) = $\frac{f'(5)g(5) - f(5)g'(5)}{2g^{2}(5) }$

Step-by-step explanation:

(a) Since k(x) is a function of two functions f(x) and g(x) [ k(x)=f(x)g(x) ], so for differentiating k(x) we need to use product rule,i.e., $\frac{\mathrm{d} [f(x)\times g(x)]}{\mathrm{d} x}=\frac{\mathrm{d} f(x)}{\mathrm{d} x}\times g(x) + f(x)\times\frac{\mathrm{d} g(x)}{\mathrm{d} x}$

this will give k'(x)=f'(x)g(x) + f(x)g'(x)

on substituting the value x=0, we will get the value of k'(0)

{for expressing the value in terms of numbers first we need to know the value of f(0), g(0), f'(0) and g'(0) in terms of numbers}{If f(0)=0 and g(0)=0, and f'(0) and g'(0) exists then k'(0)=0}

(b) m(x) is a function of two functions f(x) and g(x) [ $m(x)=\frac{1}{2}\times\frac{f(x)}{g(x)}$ ]. Since m(x) has a function g(x) in the denominator so we need to use division rule to differentiate m(x). Division rule is as follows : $\frac{\mathrm{d} \frac{f(x)}{g(x)}}{\mathrm{d} x}=\frac{\frac{\mathrm{d} f(x)}{\mathrm{d} x}\times g(x) + f(x)\times\frac{\mathrm{d} g(x)}{\mathrm{d} x}}{g^{2}(x)}$