The equation for this sequence is:

a_n = a_n-1 + 7

where n is the term's order in the sequence, a_n is the value of the term, and a_n-1 is the value of the term before a_n. This makes the next number:

a₆ = a₅ + 7
a₆ = 29 + 7
a₆ = 36

To multiply terms with radicals, multiply the coefficients and radicands separately:

7\( \sqrt{7} \) x 6\( \sqrt{9} \)
(7 x 6)\( \sqrt{7 \times 9} \)
42\( \sqrt{63} \)

Now we need to simplify the radical:

42\( \sqrt{63} \)
42\( \sqrt{7 \times 9} \)
42\( \sqrt{7 \times 3^2} \)
(42)(3)\( \sqrt{7} \)
126\( \sqrt{7} \)

The distributive property for multiplication helps in solving expressions like a(b + c). It specifies that the result of multiplying one number by the sum or difference of two numbers can be obtained by multiplying each number individually and then totaling the results: a(b + c) = ab + ac. For example, 4(10-5) = (4 x 10) - (4 x 5) = 40 - 20 = 20.

A factor is a positive integer that divides evenly into a given number. For example, the factors of 8 are 1, 2, 4, and 8.

The number of students taking German or Spanish is 15 + 14 = 29. Of that group of 29, 8 are taking both languages so they've been counted twice (once in the German group and once in the Spanish group). Subtracting them out leaves 29 - 8 = 21 who are taking at least one language. 33 - 21 = 12 students who are not taking either language.