Final answer:
The sum of the terms common to both given arithmetic series is 180, found by identifying the common terms as multiples of 12 and using the formula for the sum of an arithmetic series.
Step-by-step explanation:
The question asks to find the sum of terms common to the two given arithmetic series. The first series is 3, 6, 9, 12,... up to 78 terms, which are multiples of 3. The second series is 5, 9, 13, 17,... up to 59 terms, which are equally spaced by 4.
The common terms in both series will be the multiples of both 3 and 4, which are actually the multiples of the least common multiple of 3 and 4, which is 12. Therefore, the common terms of the series are 12, 24, 36, ..., and so on, until the last term does not exceed the highest terms in both given series.
To find the sum of the common terms, we first need to identify the last common term. The highest possible multiple of 12 that does not exceed the highest terms of the given series (78 for the first and 59 for the second series) is 60 (12 × 5). Now, we can calculate the sum of the arithmetic series whose first term (a) is 12, common difference (d) is 12, and last term (l) is 60.
The sum of an arithmetic series is given by the formula:
S = n/2 × (a + l)
where n is the number of terms.
Here, we should find the total number of terms (n) by using the formula n = (l - a)/d + 1.
So, n = (60 - 12)/12 + 1 = 5.
Now we can find the sum S:
S = 5/2 × (12 + 60) = 5/2 × 72 = 5 × 36 = 180.
Therefore, the sum of the terms common to both series is 180.