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 𝑎 and a common difference 𝑑. The first term in our series is
16. Therefore, 𝑎 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 𝑎 plus 𝑛 minus one multiplied by 𝑑. This means that the 18th term will
be 𝑎 plus 17𝑑. The 19th term would be equal to 𝑎
plus 18𝑑 and so on.
Substituting in our values of 𝑎
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 𝑎 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 𝑎
plus five 𝑑.
𝑑 is the common difference which
we know is seven. Our value for 𝑎 in this case, is
the 18th term as this is the start of the six consecutive terms. We need to substitute 𝑎 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.
