Find x: 5²⁺⁴⁺⁶⁺....⁺²ˣ = 0.04⁻²⁸ ?

First, we can get that 0.04 to be a power of 5.

0.04 = 4/100 = 1/25 = 5⁻²

So the right side is now:

0.04⁻²⁸

(5⁻²)⁻²⁸

the exponent of an exponent is the same as the product of the exponents, so:

5⁵⁶

We now have two values that are equal with the same bases, so the exponents must be equal.

The left side is the sum of an arithmetic sequence starting with 2 with a common difference of 2.  So the n'th term of that sequence is:

a(n) = a + b(n - 1)

a(n) = 2 + 2(n - 1)

a(n) = 2 + 2n - 2

a(n) = 2n

The sum of the first "n" terms is the sum of the first and last term, times the number of terms, divided by 2.

We don't know how many terms this has, but we know the sum is 56.  The last term is "2x", so we can use that to get this equation:

S(n) = [a + a(n)] n / 2

56 = [2 + 2x] n / 2

56 = 2(1 + x) n / 2

56 = (1 + x) n

If 2x is the n'th term, then:

a(n) = 2n

2x = 2n

x = n

So x and n are the same value.  We now can substitue and solve for x:

56 = (1 + x) n

56 = (1 + x) x

56 = x² + x

0 = x² + x - 56

0 = (x + 8)(x - 7)

x = -8 and 7

We can't have a negative term, so throwing that out we get:

x = 7

The left side is 5^S where S is the sum of that series.

Each term is 2n and since last term is 2x there must be x terms

The average term value is (2 + 2x)/2 = x + 1

S = average term * number of terms = x^2 + x

On the right side [0.04^(-1)]^28 = 25^28 = 5^56

Equating powers of 5, x^2 + x – 56 = 0 = (x + 8)(x - 7)

x = 7 since 7th term in that series would be 14 not -16