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Question: Let $a_1, a_2, ..., a_{2024}$ be an Arithmetic Progression such that $a_1 + (a_5 + a_{10} + a_{15} +...

Let a1,a2,...,a2024a_1, a_2, ..., a_{2024} be an Arithmetic Progression such that a1+(a5+a10+a15+...+a2020)+a2024=2233a_1 + (a_5 + a_{10} + a_{15} + ... + a_{2020}) + a_{2024} = 2233. Then a1+a2+a3+...+a2024a_1 + a_2 + a_3 + ... + a_{2024} is equal to 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A

1012

B

11132

C

2233

D

12144

Answer

11132

Explanation

Solution

The sum of an AP is Sn=n2(a1+an)S_n = \frac{n}{2}(a_1 + a_n). For n=2024n=2024, S2024=1012(a1+a2024)S_{2024} = 1012(a_1 + a_{2024}). The given equation a1+(a5+...+a2020)+a2024=2233a_1 + (a_5 + ... + a_{2020}) + a_{2024} = 2233 involves 404 terms in the parenthesis. Using the property ai+ani+1=a1+ana_i + a_{n-i+1} = a_1 + a_n, the sum of these 404 terms can be shown to be 202(a1+a2024)202(a_1 + a_{2024}). Substituting this into the given equation yields 203(a1+a2024)=2233203(a_1 + a_{2024}) = 2233, so a1+a2024=11a_1 + a_{2024} = 11. Finally, S2024=1012×11=11132S_{2024} = 1012 \times 11 = 11132.