Given that the number of calories that Karen burned each day since she started is given by sequence {349, 357, 365, 373, 381,...}
Now we have to find a rule for the given sequence.
Most of the sequences are usually AP or GP so let's test for AP.
In AP, difference between consecutive terms is always equal.
357-349=8
365-357=8
373-365=8
381-373=8
We can see that each time difference is fixed to 8.
Hence given sequence is an AP whose common difference d=8
First term a=349
Now the rule can be written using nth term formula which is
[tex] a_n=a+(n-1)d [/tex]
plug the given values
[tex] a_n=349+(n-1)*8 [/tex]
[tex] a_n=349+8n-8 [/tex]
[tex] a_n=341+8n [/tex]
Hence required rule is [tex] a_n=341+8n [/tex] where n is natural number.
Now we ahve to find the number of calories that Karen will burn on the 30th day. So plug n=30 in above rule
[tex] a_{30}=341+8*30=341+240=581 [/tex]
Hence final answer is 581 calories.
