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The sum of the first 9 terms of an arithmetic sequence is 45 and the sum of first 18 terms is 171.
(a) What is the sum of its 10th to 18th terms?
(b) What is its 5th term?
(c) Find its 14th term.
(d) Find the sum of 5th to 14th terms.

Answer
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Hint:
Here, we will substitute the given values in the formula of the sum of n terms of an AP to find two equations. Then we will use the elimination method to get the first term and common difference of the given AP. Using these values we will be able to find the answers to all the given parts.

Formula Used:
We will use the following formulas:
1. The general term of an AP is an=a+(n1)d where, a is the first term, d is the common difference and n is the total number of terms in an AP.
2. The sum of n terms of an AP is given by Sn=n2[2a+(n1)d]

Complete step by step solution:
Let the first term of an Arithmetic Progression (AP) be a, the common difference be d and the total number of terms in the AP be n
The sum of first 9 terms of an arithmetic sequence is 45.
Hence, in this situation, n=9 and S9=45
Thus, substituting these values in the formula of sum of n terms Sn=n2[2a+(n1)d], we get
S9=92[2a+(91)d]
45=92[2a+8d]
Taking 2 common from the bracket and cancelling it out with the denominator, we get
45=9[a+4d]
Dividing both sides by 9, we get
a+4d=5……………………………..(1)
Similarly, it is given that the sum of the first 18 terms is 171.
Thus, substituting n=18 and S18=171 in the formula of sum of n terms Sn=n2[2a+(n1)d], we get
S18=182[2a+(181)d]
171=9[2a+17d]
Dividing both sides by 9, we get
2a+17d=19……………………………(2)
Now, we will solve the equations (1) and (2) using elimination method.
We will multiply equation (1) by 2 and then subtract the second equation from it.
Thus, we get,
2a+8d2a17d=1019
Adding and subtracting the like terms, we get
9d=9
Dividing both sides by 9, we get
d=1
Hence, substituting this in equation (1), we get
a+4(1)=5
a=54=1
Therefore, the first term of the AP, a=1 and the common difference, d=1
Now,
(a) What is the sum of its 10th to 18th terms?
First of all, we will find the 10th term
Hence, a10=a+(101)d
Substituting the values of the first term and common difference, we get,
a10=1+9×1=1+9=10
Hence, to find the sum from 10th to 18th terms of this AP, we will consider the first term as the 10th term that is, a=10 for this case.
Also, from the 10th to 18th terms there are total 1810+1=9 terms
Hence, we will take n=9 for this case.
Thus sum of these 9 terms from 10th to 18th terms of this AP will be:
S9=92[2(10)+(91)(1)], where the common difference will remain the same
S9=92[20+8]=92×28=9×14=126
Therefore, the sum of 10th to 18th terms of this AP is 126.
(b) What is its 5th term?
Now, in order to find the 5th term of this AP, we will use the general formula, i.e.
an=a+(n1)d
Here, substituting n=5, a=1 and d=1, we get,
a5=1+(51)(1)
a5=1+4=5
Therefore, the fifth term of this AP is 5.
(c) Find its 14th term.
Similarly, like the (b) part, we will repeat the steps but in this case n=14
a14=1+(141)(1)
a14=1+13=14
Therefore, the 14th term of this AP is 14.
(d) Find the sum of 5th to 14th terms.
In this part, we will again repeat the steps of the (a) part.
Hence, in this case, we will consider the first term as a=5 because we know that the fifth term is 5.
And the total number of terms is 145+1=10
Hence, S10=102[2(5)+(101)(1)]
S10=5[10+9]=5×19=95

Therefore, the sum of 5th to 14th terms of this AP is 95.
Thus, this is the required answer.


Note:
An Arithmetic Progression is a sequence of numbers such that the difference between any term and its preceding term is constant. This difference is known as the common difference of an Arithmetic Progression (AP). A real life example of AP is when we add a fixed amount in our money bank every week. Similarly, when we ride a taxi, we pay an amount for the initial kilometer and pay a fixed amount for all the further kilometers, this also turns out to be an AP.