But is the answer unique?

$4$ average is $\boxed { 4 }$ , so it is not an answer.

$4, 44$ average is $\boxed { \frac { 4 + 44 } { 2 } \rightarrow 24 }$ , so it is not an answer even.

$4, 44, 444$ average is $\boxed { \frac { 4 + 44 + 444 } { 3 } \rightarrow 164 }$ , so it is not an answer, too.

$4, 44, 444, 4444$ average is $\boxed { \frac { 4 + 44 + 444 + 4444 } { 4 } \rightarrow 1234 }$ , so it is not the answer, no answers yet.

$4, 44, 444, 4444, 44444$ average is $\boxed { \frac { 4 + 44 + 444 + 4444 + 44444 } { 5 } \rightarrow 9876 }$ , so it is the answer.

The answer is $\boxed { 44444 } \rightarrow \boxed { 5 }$ .

A Mental approximation:

The first term = ~4

The sum of the first 2 terms = ~50, average = ~25

The sum of the first 3 terms = ~500, average = under 500

The sum of the first 4 terms = ~5000, average = ~1,250 (too low)

The sum of the first 5 terms = ~50,000, average = ~10,000 (what we want)

The sum of the first 6 terms = ~500,000, average = ~100,000 (too high)

You can check to confirm that the answer is 5