Look at the series in pairs of terms: (1,4) (7,28) (31,124)...
The second term in each pair is 4 times the first term of the pair.
The first term in each pair (apart from the first) is 3 more than the second term of the previous pair.
If we number the pairs instead of each term and we call this number n, then the first term of the pair is referred to as 2n-1 in the series. For example, 7 is the third term of the series but it is the first term of the second pair, so n=2 and the position in the series is 2n-1=3. We can write 2n as the position of the second term of the pair in the series, that is, position 4 in the series.
The first term of each pair has the value 2^(2n-1)-1. So if n=2, this value is 2³-1=8-1=7, and a₃=7 in the series, that is, the third term. a₄=4a₃=4×7=28.
31 is the first term in the third pair, so n=3 and a₅=2⁵-1=31, which obeys the rule. So a₆=4×31=124. And we can go on applying this pattern for the next pair, n=4. a₇=2⁷-1=128-1=127 and a₈=4×127=508.
Using square brackets to denote subscript we can formulate the general term for the nth pair, relating it to the position of the terms in the series. We have the first term of the pair as a[2n-1]=2^(2n-1)-1, and the second term of the pair as a[2n]=4×a[2n-1]=4(2^(2n-1)-1)=2^(2n+1)-4.
