Answer :
Final answer:
The number of terms (n) in the AP is 11 and the common difference is 12.
Explanation:
To find the number of terms (n) in the arithmetic progression (AP), we can use the formula for the sum of an AP: Sₙ = (n/2)(a + l), where Sₙ represents the cumulative total of a series, with 'a' denoting the initial term, 'l' indicating the final term, and 'n' signifying the number of terms. In this case, a = 1, l = 121, and Sₙ = 671. Plugging these values into the formula, we get: 671 = (n/2)(1 + 121). Simplifying this equation, we have: 671 = 61ₙ. When determining 'n,' we ascertain that it equals 11.
To find the common difference (d) between the terms, we can use the formula: d = (l - a)/(n - 1), where d is the common difference. Plugging in the values we already know, we have: d = (121 - 1)/(11 - 1), which simplifies to d = 12.
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Answer:
(a)11
(b)12
Step-by-step explanation:
The first term, a = 1
The last term, l=121
Sum of the series, [tex]S_n=671[/tex]
Given an arithmetic series where the first and last term is known, its sum is calculated using the formula:
[tex]S_n=\dfrac{n}{2}(a+l)[/tex]
Substituting the given values, we have:
[tex]671=\dfrac{n}{2}(1+121)\\671=\dfrac{n}{2} \times 122\\671=61n\\$Divide both sides by 61\\n=11[/tex]
(a)There are 11 terms in the arithmetic progression.
(b)We know that the 11th term is 121
The nth term of an arithmetic progression is derived using the formula:
[tex]a_n=a+(n-1)d[/tex]
[tex]a_{11}=121\\a=1\\n=11[/tex]
Therefore:
121=1+(11-1)d
121-1=10d
120=10d
d=12
The common difference between them is 12.
