Problem Study : Arithmetic Progression

Problem

An $\displaystyle A.P.$ has $\displaystyle 22$ terms. the sum of the odd terms is $\displaystyle 253$ and the sum of the even terms is $\displaystyle 275$. Find the last term.

Solution

Let $\displaystyle {{u}_{1}},{{u}_{2}},{{u}_{3}},...,{{u}_{{22}}}$ be an $\displaystyle A.P.$

Let the first term be $\displaystyle a$ and the common difference be $\displaystyle d$.

By the problem,

$\displaystyle {{u}_{1}}+{{u}_{3}}+{{u}_{5}}+...+{{u}_{{21}}}=253$

Since $\displaystyle {{u}_{1}},{{u}_{3}},{{u}_{5}},...,{{u}_{{21}}}$ is also an A.P. where the first term is $\displaystyle a$ and the common difference is $\displaystyle 2d$.

$\displaystyle \therefore$ Sum of odd terms = $\displaystyle \frac{{11}}{2}\{2a+(11-1)(2d)\}$

$\displaystyle \therefore 11(a+10d)=253$

$\displaystyle \therefore a+10d=23\  ------(1)$

Again, $\displaystyle {{u}_{2}}+{{u}_{4}}+{{u}_{6}}+...+{{u}_{{22}}}=275$

$\displaystyle {{u}_{2}},{{u}_{4}},{{u}_{6}},...,{{u}_{{22}}}$ is also an A.P. where the first term is $\displaystyle a+d$ and the common difference is $\displaystyle 2d$.

$\displaystyle \therefore$ Sum of even terms = $\displaystyle \frac{{11}}{2}\{2(a+d)+(11-1)(2d)\}$

$\displaystyle \therefore 11(a+11d)=275$

$\displaystyle \therefore a+11d=25\  ------(2)$

$\displaystyle \text{Equation}(2)-\text{Equation}(1)\Rightarrow d=2$

Substituting $\displaystyle d=2$ in equation (1),

$\displaystyle a+10(2)=23\Rightarrow a=3$.

$\displaystyle \therefore \text{Last term}={{u}_{{22}}}=a+21d=45.$

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Property I: If a constant quantity is added to or subtracted from each term of an Arithmetic Progression ($\displaystyle A.P$), then the resulting terms of the sequence are also in $\displaystyle A.P$ with same common difference ($\displaystyle d$).

Property II: If each term of an Arithmetic Progression is multiplied or divided by a non-zero constant quantity, then the resulting sequence form an Arithmetic Progression.

Property III: In an Arithmetic Progression of finite number of terms the sum of any two terms equidistant from the beginning and the end is equal to the sum of the first and last terms.

Property IV: A sequence is an Arithmetic Progression if and only if the sum of its first n terms is of the form $\displaystyle A{{n}^{2}}+Bn$, where $\displaystyle A,B$ are two constant quantities that are independent of $\displaystyle n$.

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