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If a, b, c are in A.P., then a+c=?

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Last updated date: 04th Mar 2024
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IVSAT 2024
Answer
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Hint: For the question of this kind where a, b, c are in arithmetic progression then find the value of the a+ c.
In solving this question we must take the first term as a and the other terms as b, c as given in the question and proceed further calculation using the arithmetic progression property of common difference is the same and solve it.

Complete step by step answer:
Firstly, let us consider the first term as \[{{t}_{1}}=a\] and the next term as \[{{t}_{2}}=b\] and the next term that is the third term as \[{{t}_{3}}=c\].
Here by using the arithmetic progression property which is the common difference from the successive terms is the same that is if \[{{t}_{1}},{{t}_{2}},{{t}_{3}}\] are successive terms in arithmetic progression then we can write\[\Rightarrow {{t}_{2}}={{t}_{1}}+d\]and also \[\Rightarrow {{t}_{3}}={{t}_{1}}+2d\] which is d using that we must conclude that the second term and the third term can be rewritten as follows.
\[\Rightarrow {{t}_{1}}=a\]
From using the property above mentioned we can rewrite the value of b as follows.
\[\Rightarrow {{t}_{2}}=a+d=b\]
By taking the help of the property of arithmetic progression we can rewrite the value of c as follows.
\[\Rightarrow {{t}_{3}}=a+2d=c\]
After rewriting the values of the a, b, c which we got using the arithmetic progression property that is the common difference is the same for the successive terms we will add the values of a and c .
\[\Rightarrow a+c=a+\left( a+2d \right)\]
Taking common the values we got the equation will be reduced as follows.
\[\Rightarrow a+c=2\left( a+d \right)\]
Here we substitute the value of a+ d which is b. So the equation will be reduced as follows.
\[\Rightarrow a+c=2b\]

So the solution of the given question will be \[a+c=2b\].

Note: We must be very careful in doing the calculations and one must be knowing the concept of arithmetic progression and must use the basic property of common difference is same for successive terms.
Students must be very careful in taking the common difference for the next successive terms and must not forget and must not think that difference is the same for any terms.
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