Expand (5 + z)4
choices are:

625 + 500z + 150z2 + 20z3 + z4

625 + 375z + 150z2 + 15z3 + z4

625 + 250z + 100z2 + 10z3 + z4

625 + 125z + 25z2 + 5z3 + z4

A

Step-by-step explanation:

We want to expand: . We need to use the Binomial Theorem for this, which states that for a monic expression like x + k (where k is a constant) to the nth power, the coefficients of the terms will follow the terms in the nth row of the Pascal Triangle. These terms in the Pascal Triangle are of the form:

nC0 , nC1 , nC2 , .... , nC(n - 1) , nCn    ----  here, nCk is combinations: , where ! denotes factorial

So, since the power here is 4, we will be using: 4C0, 4C1, 4C2, 4C3, and 4C4 as our coefficients of the terms.

Let's make 5 + z into z + 5. Now, we expand:

4C0 = 1 , 4C1 = 4 , 4C2 = 6 , 4C3 = 4 , and 4C4 = 1:

Thus, the answer is A.

Hope this helps!

Option 1

625 + 500z + 150z² + 20z³ + z⁴

Step-by-step explanation:

(5 + z)⁴

4C0×5⁰×z⁴ + 4C1×5¹×z³ + 4C2×5²×z² + 4C3×5³×z¹ + 4C4×5⁴×z⁰

z⁴ + 20z³ + 150z² + 500z + 625

, and

for

step-by-step explanation:

we can analyze the pattern at which the terms increase. let us denote , , , so that we can call

note the following:

the pattern: to get the next term value, we just add the next term number (e.g. 1st, 2nd, to the previous term value (term values are the given numbers in the question, 1, 3, 6, 10, 15,

from , to get to , we add

from , to get to , we add

etc.

we need to form a recursive formula, a formula that has the next term as a function of the previous term. so a possible recursive formula can be

, and for