Answer:
(b) 3
Step-by-step explanation:
You want the common difference of an arithmetic progression that has 5th term 20 and 64 as the sum of the 7th and 11th terms.
The sum of the 7th and 11 terms will be twice the value of the term halfway between: the 9th term. That means the 9th term is 64/2 = 32.
The 9th term and the 5th term differ by (9-5)=4 times the common difference:
4d = 32 -20 = 12
d = 12/4 = 3
The common difference of the sequence is 3, choice B.
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Additional comment
The above represents an approach that can be executed with mental arithmetic. More formally, we could start with the definition of the n-th term, and solve the resulting equations.
an = a1 +d(n -1)
a5 = a1 +d(5 -1) ⇒ 20 = a1 +4d
a7 +a11 = (a1 +d(7 -1)) +(a1 +d(11 -1)) ⇒ 64 = 2a1 +16d
Subtracting twice the first equation from the second gives an equation with the a1 terms canceled:
-2(20) +(64) = -2(a1 +4d) +(2a1 +16d)
24 = 8d
3 = d
The attachment shows the first 11 terms of the progression.
Answer:
[tex](b)3[/tex]
Step-by-step explanation:
Let's analyze the information:
The 5th term is 20, so the AP is 1, 4, 7, 10, 13, ... (starting from 1).
The sum of the 7th and 11th terms is 64. The 7th term is 2(7) + 1 = 15, and the 11th term is 2(11) + 1 = 23. Thus, 15 + 23 = 64.
Now, consider the common difference (d):
If d = 4, the AP becomes 1, 5, 9, 13, 17, ..., which doesn't match the given terms.
If d = 3, the AP becomes 1, 4, 7, 10, 13, ..., which matches the given terms.
So, the common difference is (b) 3.
