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Questions
In an AP given a = 8, an = 62, Sn = 210, find n and d.
Let there be an A.P. with the first term 'a', common difference 'd'. If an a denotes in nth term and Sn the sum of first n terms, find.
n and d, if a = 8, an = 62 and Sn = 210
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Solution 1
a = 8, an = 62 and Sn = 210 (Given)
∵ Sn = `"n"/2` [a + an]
⇒ 210 = `"n"/2 [8 + 62]`
⇒ 210 = `"n"/2` × 70
⇒ 35n = 210
⇒ n = `210/35`
⇒ n = 6
∵ an = a + (n - 1) × d
⇒ 62 = 8 + (6 - 1) × d
⇒ 62 = 8 + 5d
⇒ 5d = 62 - 8
⇒ 5d = 54
⇒ d = `54/5`
Hence, the required values of n and 4 are 6 and `54/5` respectively.
Solution 2
Here, we have an A.P. whose nth term (an), the sum of first n terms (Sn) and first term (a) are given. We need to find the number of terms (n) and the common difference (d).
Here,
First term (a) = 8
Last term (`a_n`) = 62
Sum of n terms (Sn) = 210
Now, here, the sum of the n terms is given by the formula,
`S_n = (n/2)(a + l)`
Where a is the first term
l = the last term
So, for the given A.P, on substituting the values in the formula for the sum of n terms of an A.P., we get,
`210 = (n/2)[8 + 62]`
210(2) = n(70)
`n = 420/70`
n = 6
Also, here we will find the value of d using the formula,
an = a + (n - 1)d
So, substituting the values in the above mentioned formula
62 = 8 + (6 - 1)d
62 - 8 = (5)d
`54/5 = d`
`d = 54/5`
Therefore, for the given A.P `n = 6 and d = 54/5`.
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