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Question Video: Finding the Value of a Certain Term of a Geometric Sequence given the Sum of Its n Terms Mathematics • 10th Grade

Find the fourth term of the geometric sequence given by 𝑆_(𝑛) = 1024 βˆ’ 2^(10 βˆ’ 𝑛) where 𝑆_(𝑛) is the sum of the first 𝑛 terms.

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Video Transcript

Find the fourth term of the geometric sequence given by 𝑆 of 𝑛 is equal to 1024 minus two to the power of 10 minus 𝑛 where 𝑆 of 𝑛 is the sum of the first 𝑛 terms.

We can start this question by calculating the sum of the first four terms of the geometric sequence. When 𝑛 is equal to four, 𝑆 of four is equal to 1024 minus two to the power of 10 minus four. 10 minus four is equal to six, and two to the power of six is equal to 64. We need to subtract this from 1024. This gives us 960. The sum of the first four terms is 960.

We want to calculate the fourth term of the sequence. This will be equal to the sum of the first four terms minus the sum of the first three terms. 𝑆 of three will be equal to 1024 minus two to the power of 10 minus three. As 10 minus three is equal to seven, we need to calculate two to the power of seven. This is equal to 128. Subtracting this from 1024 gives us 896, which is the sum of the first three terms.

We can now calculate the fourth term of the sequence by subtracting 896 from 960. This is equal to 64. The fourth term of the geometric sequence given by 𝑆 of 𝑛 is equal to 1024 minus two to the power of 10 minus 𝑛 is 64.

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