
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 to terms?
(b) What is its term?
(c) Find its term.
(d) Find the sum of to terms.
Answer
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Hint:
Here, we will substitute the given values in the formula of the sum of 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 where, is the first term, is the common difference and is the total number of terms in an AP.
2. The sum of terms of an AP is given by
Complete step by step solution:
Let the first term of an Arithmetic Progression (AP) be , the common difference be and the total number of terms in the AP be
The sum of first 9 terms of an arithmetic sequence is 45.
Hence, in this situation, and
Thus, substituting these values in the formula of sum of terms , we get
Taking 2 common from the bracket and cancelling it out with the denominator, we get
Dividing both sides by 9, we get
……………………………..
Similarly, it is given that the sum of the first 18 terms is 171.
Thus, substituting and in the formula of sum of terms , we get
Dividing both sides by 9, we get
……………………………
Now, we will solve the equations and using elimination method.
We will multiply equation by 2 and then subtract the second equation from it.
Thus, we get,
Adding and subtracting the like terms, we get
Dividing both sides by , we get
Hence, substituting this in equation , we get
Therefore, the first term of the AP, and the common difference,
Now,
(a) What is the sum of its to terms?
First of all, we will find the term
Hence,
Substituting the values of the first term and common difference, we get,
Hence, to find the sum from to terms of this AP, we will consider the first term as the term that is, for this case.
Also, from the to terms there are total terms
Hence, we will take for this case.
Thus sum of these 9 terms from to terms of this AP will be:
, where the common difference will remain the same
Therefore, the sum of to terms of this AP is 126.
(b) What is its term?
Now, in order to find the term of this AP, we will use the general formula, i.e.
Here, substituting , and , we get,
Therefore, the fifth term of this AP is 5.
(c) Find its term.
Similarly, like the (b) part, we will repeat the steps but in this case
Therefore, the term of this AP is 14.
(d) Find the sum of to 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 because we know that the fifth term is 5.
And the total number of terms is
Hence,
Therefore, the sum of to 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.
Here, we will substitute the given values in the formula of the sum of
Formula Used:
We will use the following formulas:
1. The general term of an AP is
2. The sum of
Complete step by step solution:
Let the first term of an Arithmetic Progression (AP) be
The sum of first 9 terms of an arithmetic sequence is 45.
Hence, in this situation,
Thus, substituting these values in the formula of sum of
Taking 2 common from the bracket and cancelling it out with the denominator, we get
Dividing both sides by 9, we get
Similarly, it is given that the sum of the first 18 terms is 171.
Thus, substituting
Dividing both sides by 9, we get
Now, we will solve the equations
We will multiply equation
Thus, we get,
Adding and subtracting the like terms, we get
Dividing both sides by
Hence, substituting this in equation
Therefore, the first term of the AP,
Now,
(a) What is the sum of its
First of all, we will find the
Hence,
Substituting the values of the first term and common difference, we get,
Hence, to find the sum from
Also, from the
Hence, we will take
Thus sum of these 9 terms from
Therefore, the sum of
(b) What is its
Now, in order to find the
Here, substituting
Therefore, the fifth term of this AP is 5.
(c) Find its
Similarly, like the (b) part, we will repeat the steps but in this case
Therefore, the
(d) Find the sum of
In this part, we will again repeat the steps of the (a) part.
Hence, in this case, we will consider the first term as
And the total number of terms is
Hence,
Therefore, the sum of
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.
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