Since this is an arithmetic sequence, each entry in the sequence is obtained by adding the same number to the previous entry - this number is

$\displaystyle 107-90 = 17$

Let $\displaystyle n$ be the number in the square. Then 

$\displaystyle n + 3 \cdot 17 = 90$

$\displaystyle n + 51 = 90$

$\displaystyle n + 51 - 51 = 90 - 51$

$\displaystyle n = 39$

An arithmetic sequence is formed by adding the same expression to each term to get the next term; this common difference is 

$\displaystyle G - 5$.

To obtain the fifth term, add $\displaystyle G - 5$ to the second term three times - equivalently, add three times this to the second term;

$\displaystyle G + 3 (G - 5) = G + 3G - 15 = 4G -15$

Since this is a geometric sequence, each entry in the sequence is obtained by multiplying the previous entry by the same number  - this number is

$\displaystyle 35 \div 5 = 7$.

Now we can find the next three entries in the sequence:

$\displaystyle 35 \times 7 = 245$  

This replaces the square.

$\displaystyle 245 \times 7 = 1,715$ 

$\displaystyle 1,715$ replaces the triangle.

$\displaystyle 1,715\times 7 = 12,005$  

$\displaystyle 12,005$ replaces the circle and is therefore the correct answer.

Since this is an arithmetic sequence, each entry in the sequence is obtained by adding the same number to the previous entry - this number is

$\displaystyle 35- 5 = 30$.

The next three entries in the sequence are computed as follows:

$\displaystyle 35 + 30 = 65$, which replaces the square

$\displaystyle 65 + 30 = 95$, which replaces the triangle

$\displaystyle 95 + 30 = 125$, which replaces the circle

Each term of a geometric sequence is obtained by multiplying the previous one by the same number (common ratio); this number is

$\displaystyle 8000 \div 400 = 20$.

Let $\displaystyle s$ be the number in the square. 

$\displaystyle s \cdot 20 ^{3} = 400$

$\displaystyle s \cdot 8000 = 400$

$\displaystyle s \cdot 8000 \div 8000= 400 \div 8000$

$\displaystyle s = 0.05$

To express $\displaystyle F_{1,000}$, the one-thousandth term of the sequence, in terms of $\displaystyle F_{997}$ and $\displaystyle F_{998}$ alone, we note that, by definition of the sequence, each term, except for the first two, is equal to the sum of the previous two. Therefore,

$\displaystyle F_{999} = F_{997}+ F_{998}$

Also

$\displaystyle F_{1,000} = F_{998}+ F_{999}$, and, substituting:

$\displaystyle F_{1,000} = F_{998}+ \left ( F_{997}+ F_{998} \right )$

and

$\displaystyle F_{1,000} = F_{997}+ 2 F_{998}$,

the correct choice.

The Fibonacci sequence begins as follows:

$\displaystyle 1,1,2,3,5,8,13,...$

This sequence is seen to be an increasing sequence. Therefore, each term is greater than its preceding term. In particular, 

$\displaystyle F_{50 }> F_{49}$

If we substitute 51 for $\displaystyle n$ in the rule of the sequence, we get

$\displaystyle F_{n} = F_{n-1}+ F_{n-2}$

$\displaystyle F_{51} = F_{51-1}+ F_{51-2}$

$\displaystyle F_{51} = F_{50}+ F_{49}$

$\displaystyle F_{50 }> F_{49}$, so

$\displaystyle F_{50 }+ F_{50} > F_{49} + F_{50 }$

$\displaystyle 2 \cdot F_{50} > F_{51 }$

This makes (a) greater.

Setting $\displaystyle n = 1,000$:

$\displaystyle a_{n} = 3a_{n-1} - a_{n-2}$

$\displaystyle a_{1,000} = 3a_{1,000-1} - a_{1,000-2}$

$\displaystyle a_{1,000} = 3a_{999} - a_{998}$

Similarly,

$\displaystyle a_{999} = 3a_{998 } - a_{997}$

Substituting:

$\displaystyle a_{1,000} = 3 (3a_{998 } - a_{997}) - a_{998}$

$\displaystyle =9a_{998 } - 3 a_{997} - a_{998}$

$\displaystyle =8a_{998 } - 3 a_{997}$