Question:
For a Fibonacci sequence, from the third term onwards, each term in the sequence is the sum of the previous two terms in that sequence. If the difference in squares of 7th and 6th terms of this sequence is 517, what is the 10th term of this sequence? It is known that all the terms of the Fibonacci sequence are natural numbers
For a Fibonacci sequence, from the third term onwards, each term in the sequence is the sum of the previous two terms in that sequence. If the difference in squares of 7th and 6th terms of this sequence is 517, what is the 10th term of this sequence? It is known that all the terms of the Fibonacci sequence are natural numbers
Show Hint
Show that \(t_7^2-t_6^2=t_5\,t_8\) and factorise 517 as \(11\times 47\).
Updated On: Nov 4, 2026
- 123
- 76
- 147
- Cannot be determined
Show Solution
Verified By Collegedunia
The Correct Option is A
Solution and Explanation
Step 1: Factorise the difference of squares.
Let the terms be \(t_1,t_2,\dots\) with \(t_{k+2}=t_{k+1}+t_k\). Then \(t_7-t_6=t_5\) and \(t_7+t_6=t_8\), so
\[t_7^2-t_6^2=(t_7-t_6)(t_7+t_6)=t_5\,t_8=517=11\times 47.\]Step 2: Choose the factor pair.
\(517=1\times 517=11\times 47\). With \(t_1,t_2\ge 1\), \(t_5=2t_1+3t_2\ge 5\), and \(t_5<t_8\). So \(t_5=11\) and \(t_8=47\) (the pair 1 and 517 is impossible because \(t_5\ge 5\)).
Step 3: Find the first two terms.
With \(t_1=a,\ t_2=b\): \(t_5=2a+3b=11\) and \(t_8=8a+13b=47\). From the first, \((a,b)=(4,1)\) or \((1,3)\). Only \((1,3)\) satisfies the second: \(8+39=47\).
Step 4: Write the sequence.
1, 3, 4, 7, 11, 18, 29, 47, 76, 123. Check: \(29^2-18^2=841-324=517\).
Final Answer:
The 10th term is 123, option (1).
Let the terms be \(t_1,t_2,\dots\) with \(t_{k+2}=t_{k+1}+t_k\). Then \(t_7-t_6=t_5\) and \(t_7+t_6=t_8\), so
\[t_7^2-t_6^2=(t_7-t_6)(t_7+t_6)=t_5\,t_8=517=11\times 47.\]Step 2: Choose the factor pair.
\(517=1\times 517=11\times 47\). With \(t_1,t_2\ge 1\), \(t_5=2t_1+3t_2\ge 5\), and \(t_5<t_8\). So \(t_5=11\) and \(t_8=47\) (the pair 1 and 517 is impossible because \(t_5\ge 5\)).
Step 3: Find the first two terms.
With \(t_1=a,\ t_2=b\): \(t_5=2a+3b=11\) and \(t_8=8a+13b=47\). From the first, \((a,b)=(4,1)\) or \((1,3)\). Only \((1,3)\) satisfies the second: \(8+39=47\).
Step 4: Write the sequence.
1, 3, 4, 7, 11, 18, 29, 47, 76, 123. Check: \(29^2-18^2=841-324=517\).
Final Answer:
The 10th term is 123, option (1).
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