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Blababa [14]
3 years ago
13

Let a4+a8+a12+a16=224. Find the sum of first 19 term in sequence.

Mathematics
1 answer:
Tomtit [17]3 years ago
7 0

Without knowing anything about the sequence, this is impossible to answer. But suppose the sequence is arithmetic, in which case each term differs by some constant k:

a_2=a_1+k

a_3=a_2+k=a_1+2k

a_4=a_3+k=a_1+3k

\ldots

a_n=a_{n-1}+k=a_1+(n-1)k

Then we can write

a_4+a_8+a_{12}+a_{16}=4a_1+36k=224\implies a_1+9k=56

and from the formula above, we see this means the 10th term in the sequence is a_{10}=56. But that's all the specific info we can gather about such an arithmetic sequence. If we set the first term to be some unknown a_1=a, then the sum of the first 19 terms in the sequence would be

\displaystyle\sum_{n=1}^{19}a_n=\sum_{n=1}^{19}\big(a+(n-1)k\big)=19a+k\sum_{n=1}^{18}n=19a+171k

If we knew one more term in the sequence, we could determine the value of k and derive the value of a (if the first term a_1 is not immediately given), and then go on to find an exact numeric value for the sum.

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The complete question in the attached figure

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