Question

Let there be an A.P. with first term 'a', common difference 'd'. If a_n denotes in n^th term and S_n the sum of first n terms, find.

(i) n and S_n, if a = 5, d = 3 and a_n = 50.

(ii) n and a, if a_n = 4, d = 2 and S_n = −14.

(iii) d, if a = 3, n = 8 and S_n = 192.

(iv) a, if a_n = 28, S_n = 144 and n= 9.

(v) n and d, if a = 8, a_n = 62 and S_n = 210

(vi) n and a_n, if a= 2, d = 8 and S_n = 90.

(vii) $k,\mathrm{if}{S}_n=3n^2+5n\mathrm{and}{a}_k=164$.

(viii) $S_{22},\mathrm{if}d=22\mathrm{and}{a}_{22}=149$

Answer 1

(i) Here, we have an A.P. whose n^th term (a_n), first term (a) and common difference (d) are given. We need to find the number of terms (n) and the sum of first n terms (S_n).

Here,

First term (a) = 5

Last term () = 50

Common difference (d) = 3

So here we will find the value of n using the formula,

So, substituting the values in the above mentioned formula

Further simplifying for n,

Now, here we can find the sum of the n terms of the given A.P., using the formula,

Where, a = 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,

Therefore, for the given A.P

(ii) Here, we have an A.P. whose n^th term (a_n), sum of first n terms (S_n) and common difference (d) are given. We need to find the number of terms (n) and the first term (a).

Here,

Last term () = 4

Common difference (d) = 2

Sum of n terms (S_n) = −14

So here we will find the value of n using the formula,

So, substituting the values in the above mentioned formula

Now, here the sum of the n terms is given by the formula,

Where, a = 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,

Equating (1) and (2), we get,

So, we get the following quadratic equation,

Further, solving it for a by splitting the middle term,

So, we get,

Or

Substituting, in (1),

Here, we get n as negative, which is not possible. So, we take,

Therefore, for the given A.P

(iii) Here, we have an A.P. whose first term (a), sum of first n terms (S_n) and the number of terms (n) are given. We need to find common difference (d).

Here,

First term () = 3

Sum of n terms (S_n) = 192

Number of terms (n) = 8

So here we will find the value of n using the formula,

So, to find the common difference of this A.P., we use the following formula for the sum

of n terms of an A.P

Where; a = first term for the given A.P.

d = common difference of the given A.P.

n = number of terms

So, using the formula for n = 8, we get,

Further solving for d,

Therefore, the common difference of the given A.P. is.

(iv) Here, we have an A.P. whose n^th term (a_n), sum of first n terms (S_n) and the number of terms (n) are given. We need to find first term (a).

Here,

Last term () = 28

Sum of n terms (S_n) = 144

Number of terms (n) = 9

Now,

Also, using the following formula for the sum of n terms of an A.P

Where; a = first term for the given A.P.

d = common difference of the given A.P.

n = number of terms

So, using the formula for n = 9, we get,

Multiplying (1) by 9, we get

Further, subtracting (3) from (2), we get

Therefore, the first term of the given A.P. is.

(v) Here, we have an A.P. whose n^th term (a_n), sum of first n terms (S_n) and first term (a) are given. We need to find the number of terms (n) and the common difference (d).

Here,

First term () = 8

Last term () = 62

Sum of n terms (S_n) = 210

Now, here the sum of the n terms is given by the formula,

Where, a = 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,

Also, here we will find the value of d using the formula,

So, substituting the values in the above mentioned formula

Therefore, for the given A.P

(vi) Here, we have an A.P. whose first term (a), common difference (d) and sum of first n terms are given. We need to find the number of terms (n) and the n^th term (a_n).

Here,

First term (a) = 2

Sum of first n^th terms () = 90

Common difference (d) = 8

So, to find the number of terms (n) of this A.P., we use the following formula for the sum

of n terms of an A.P

Where; a = first term for the given A.P.

d = common difference of the given A.P.

n = number of terms

So, using the formula for n = 8, we get,

Further solving the above quadratic equation,

Further solving for n,

Now,

Also,

Since n cannot be a fraction

Thus, n = 5

Also, we will find the value of the n^th term (a_n) using the formula,

So, substituting the values in the above mentioned formula

Therefore, for the given A.P.

(vii)
$$\begin{gathered} a_k=S_k-S_{k-1} \\ \Rightarrow 164=\left(3k^2+5k\right)-\left(3{\left(k-1\right)}^2+5\left(k-1\right)\right) \\ \Rightarrow 164=3k^2+5k-3k^2+6k-3-5k+5 \\ \Rightarrow 164=6k+2 \\ \Rightarrow 6k=162 \\ \Rightarrow k=27 \end{gathered}$$

(viii) Given d = 22, $a_{22}=149$ ,n=22
We know that

$$\begin{gathered} 149=a+(22-1)22 \\ 149=a+462 \\ a=-313 \end{gathered}$$

Now, Sum is given by

Where; a = first term for the given A.P.

d = common difference of the given A.P.

n = number of terms

So, using the formula for n = 22, we get

$$\begin{gathered} S_{22}=\frac{22}{2}\left\{2\times \left(-313)+(22-1)\times 22\right)\right\} \\ {S}_{22}=11\left\{-626+462\right\} \\ {S}_{22}=-1804 \end{gathered}$$

Hence, the sum of 22 terms is −1804.