The first term of a sequence is 2021. Each succeeding term is the sum of the squares of the digits of the previous term. What is the 2021th term in the sequence?
Hint: If at first you do not see a pattern, keep going!
Options:
4
9
16
25
36
49
64
Given :– The first term of a sequence is 2021. Each succeeding term is the sum of the squares of the digits of the previous term. What is the 2021th term in the sequence ? Hint: If at first you do not see a pattern, keep going !
Answer :–
→ First term = 2021
→ second term = 2² + 0² + 2² + 1 = 4 + 0 + 4 + 1 = 9
→ third term = 9² = 81
→ fourth term = 8² + 1² = 64 + 1 = 65
→ fifth term = 6² + 5² = 36 + 25 = 61
→ sixth term = 6² + 1² = 36 + 1 = 37
→ 7th ,, = 3² + 7² = 9 + 49 = 58
→ 8th ,, = 5² + 8² = 25 + 64 = 89
→ 9th = 8² + 9² = 64 + 81 = 145
→ 10th = 1² + 4² + 5² = 1 + 16 + 25 = 42
→ 11th = 4² + 2² = 16 + 4 = 20
→ 12th = 2² + 0² = 4
→ 13th = 4² = 16
→ 14th = 1² + 6² = 37
conclusion :-
→ In 2016th terms = 8th term will repeat .
so,
→ total pairs = 2016/8 = 252 = no remainder .
therefore, we can conclude that, the 2021th term of the sequence is 37 .
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Given : The first term of a sequence is 2021. Each succeeding term is the sum of the squares of the digits of the previous term.
To Find : 2021 th term in the sequence
4
9
16
25
36
49
64
Solution:
1st term = 2021
2nd term = 2² + 0² + 2² + 1 = 4 + 0 + 4 + 1 = 9
3rd term = 9² = 81
4th term = 8² + 1² = 64 + 1 = 65
5th term = 6² + 5² = 36 + 25 = 61
6th term = 6² + 1² = 36 + 1 = 37
7th term = 3² + 7² = 9 + 49 = 58
8th term = 5² + 8² = 25 + 64 = 89
9th term = 8² + 9² = 64 + 81 = 145
10th term = 1² + 4² + 5² = 1 + 16 + 25 = 42
11th term = 4² + 2² = 16 + 4 = 20
12th term = 2² + 0² = 4
13th term = 4² = 16
14th term = 1² + 6² = 37
Sequence will repeat now
6th , 14th , 22 nd would be same
6th , 14th , 22nd — 37 8k + 6 th term k is positive integer
7th , 17th , 23rd — 58 8k + 7 th term
8th , 16th , 24th – 89 8k th term
9th , 17th , 25 th 145 8k +1 term
10th , 18th , 26th 42 8k + 2 term
11th , 19th , 27th 20 8k + 3 term
12th , 20th , 28th 4 8k + 4 term
13th , 21st , 29th = 16 8k + 5th term
2021 = 252 * 8 + 5
=8k + 5th term
Hence 2021th term in the sequence is 16
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