The first four terms of a sequence are 1, 4, 2, and 3. Beginning with the fifth term in the sequence, each term is the sum of the previous four terms. Therefore, the fifth term is 10. What is the eighth term?
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The first four terms of a sequence are 1, 4, 2, and 3. Beginning with the fifth term in the sequence, each term is the sum of the previous four terms. Therefore, the fifth term is 10. What is the eighth term?
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Answer:
Class 11
>>Applied Mathematics
>>Sequences and series
>>Arithmetic progression
>>(a) Write the algebraic form of the arit
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(a) Write the algebraic form of the arithmetic sequence 1,4,7,10,..
(b) Is 100 a term of this sequence? Why?
(c) Prove that the square of any term of this sequence is also a term of it.
Medium
Updated on : 2022-09-05
Solution
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Sequence is 1,4,7,10,.....
(a) a=1,d=4−1=3
t
n
=a+(n−1)d
⇒t
n
=1+(n−1)d
⇒t
n
=1+3n−3
⇒t
n
=3n−2 which is algebraic form of the given arithmetic sequence.
(b) t
n
=3n−2
⇒t
n
=3n−2
⇒3n=102
⇒n=34
⇒100 is the 34
th
term of the sequence.
(c) We know, t
n
=3n−2
Square of the term of this sequence
=(3n−2)
2
=9n
2
−12n+4
=9n
2
−12n+12−8
=3[3n−2)
2
−2]−2
Let k=[(3n−2)
2
−2]
Square of the term of this sequence =3k−2
Since square of the sequence is of the form =3k−2, therefore square of any term of the sequence is a term of this sequence.