Find the sum of the first five terms of the geometric progression if a2 = 4, a3 = 7.

Recall the formula for finding the sum of the first n terms of an arithmetic progression (Sn):

Sn = n * (a1 + an) / 2, where n is the number of added members of the progression, a1 is the first member of the progression, an is the n-th member of the progression.

This means that in order to find S5, you need to know a1 and a5.

Knowing the second and third terms of the arithmetic progression, you can find its difference (d) by the formula:

d = an + 1 – an.

Let’s calculate the difference of the given progression:

d = a3 – a2 = 7 – 4 = 3.

Find the first term of the progression by expressing its value from the formula d = an + 1 – an.

an = an + 1 – d.

a1 = a2 – d,

a1 = 4 – 3 = 1.

Let’s find the fifth term of the progression.

Let’s recall the formula for finding the n-th term of the arithmetic progression (an):

an = a1 + (n – 1) * d, where n is the ordinal number of the sought-for member of the progression, d is the difference of the progression.

a5 = 1 + (5 – 1) * 3 = 1 + 4 * 3 = 1 + 12 = 13.

Now let’s calculate the sum of the first five terms of the given progression:

S5 = 5 * (1 + 13) / 2 = 5 * 14/2 = 35.

Answer: 35.