the first term of a linear sequence is 3 and 8th term is 31 .find the common difference and hence find the 20th term
since the two terms are 7 terms apart, 7d = 31-3 = 28
So, now you know d, you can find a, and then
a + 19d.
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To find the common difference in a linear sequence, we need to find the difference between any two consecutive terms.
Given:
First term (a₁) = 3
Eighth term (a₈) = 31
We can use the formula for the nth term of an arithmetic sequence to find the common difference:
aₙ = a₁ + (n-1)d,
where aₙ is the nth term, a₁ is the first term, n is the position of the term in the sequence (in this case, 8), and d is the common difference.
Substituting the values we know:
31 = 3 + (8-1)d
Simplifying the equation:
31 = 3 + 7d
Subtracting 3 from both sides:
28 = 7d
Dividing both sides by 7:
d = 4
Therefore, the common difference is 4.
To find the 20th term, we can again use the formula for the nth term of an arithmetic sequence:
aₙ = a₁ + (n-1)d
Substituting the values we know:
a₂₀ = 3 + (20-1)4
Simplifying the equation:
a₂₀ = 3 + 19*4
Calculating:
a₂₀ = 3 + 76
a₂₀ = 79
Therefore, the 20th term of the sequence is 79.