Video: Solving Word Problems Involving Arithmetic Series

Find the sum of the 6 consecutive terms that start from the eighteenth term of the arithmetic series 16 + 23 + 30 + β‹…β‹…β‹….

04:11

Video Transcript

Find the sum of the six consecutive terms that start from the 18th term of the arithmetic series 16 plus 23 plus 30 and so on.

Any arithmetic series has a first term π‘Ž and a common difference 𝑑. The first term in our series is 16. Therefore, π‘Ž is equal to 16. The difference between each of the terms, 16 and 23 and 23 and 30, is seven. Therefore, 𝑑 is equal to seven. We’re asked to find the sum of the six consecutive terms from the 18th term in this sequence. One way of doing this would be to work out the 18th, 19th, 20th, 21st, 22nd, and 23rd terms. We can then find the sum of them by adding these six numbers. The 𝑛th term of any arithmetic series can be found using the formula π‘Ž plus 𝑛 minus one multiplied by 𝑑. This means that the 18th term will be π‘Ž plus 17𝑑. The 19th term would be equal to π‘Ž plus 18𝑑 and so on.

Substituting in our values of π‘Ž and 𝑑 gives us 16 plus 17 multiplied by seven. This is equal to 135. The 18th term in the series is 135. As the common difference is seven, the 19th term will be 142. We could check this by calculating 16 plus 18 multiplied by seven; this is 142. The 20th term is equal to 149, the 21st is 156, the 22nd is 163, and the 23rd is 170. We can then calculate the sum by adding these six numbers. This gives us an answer of 915. The sum of the six consecutive terms that start from the 18th term of the arithmetic series is 915.

An alternative method, once we have found the 18th term, is to use the formula 𝑆 of 𝑛 is equal to 𝑛 divided by two multiplied by two π‘Ž plus 𝑛 minus one multiplied by 𝑑. This enables us to calculate the sum of any consecutive terms. In this question, we need to calculate the sum of six consecutive terms. Therefore, we need to calculate 𝑆 of six substituting in 𝑛 equal six gives us sex divided by two multiplied by two π‘Ž plus five 𝑑.

𝑑 is the common difference which we know is seven. Our value for π‘Ž in this case, is the 18th term as this is the start of the six consecutive terms. We need to substitute π‘Ž equals 135. Six divided by two is equal to three. So we’re left with three multiplied by two multiplied by 135 plus five multiplied by seven. Two times 135 is 270. Five times seven is equal to 35. Adding this to 270 gives us 305. Finally, three multiplied by 305 is 915. We can, therefore, conclude that our answer of 915 was correct.

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.