Evaluate the sum: 2 + 2^2 + 2^3 + … + 2^9.

Given data

• Geometric series of powers of 2: 2 + 2^2 + 2^3 + … + 2^9.
• Notation note: The original text '22, 23, …, 29' is interpreted as 2^2, 2^3, …, 2^9 (consistent with typical formatting issues in plain text). This preserves all numbers and the crux.

Concept / Approach

• For a geometric series a + ar + ar^2 + … + ar^{n-1}, the sum is S = a(1 − r^n)/(1 − r) for r ≠ 1.
• Here a = 2, r = 2, and terms run from exponent 1 to 9 (9 terms).

Step-by-step calculation

S = 2 + 2^2 + … + 2^9 = (2^1 + 2^2 + … + 2^9)= \u2211_{k=1}^{9} 2^k = (2^{10} − 2^1)= 1024 − 2 = 1022

Verification / Alternative

Direct check (partial sums): 2 + 4 = 6; +8 = 14; +16 = 30; +32 = 62; +64 = 126; +128 = 254; +256 = 510; +512 = 1022.

Common pitfalls

• Including 2^0 = 1 by mistake (that would give 2^{10} − 1 = 1023).
• Misreading the plain-text exponents as multi-digit numbers.

Final Answer

1022