Question

Choose the correct alternative to complete the series: $11,32,53,74,95,$?
A. $106$
B. $111$
C. $113$
D. $116$

Answer 1

Hint: The series $11,32,53,74,95,$is an Arithmetic Progression (AP). So, using the formula of Arithmetic Progression, we find the next number of the series. The formula of AP is ${a_n} = {a_1} + (n - 1)d$ where ${a_n}$ is the ${n^{th}}$ element, ${a_1}$ is the first element and $d$ is the common difference between two consecutive terms.

Complete step-by-step solution:
The given series is $11,32,53,74,95,$with first element ${a_1} = 11$, common difference \[d = {a_2} - {a_1} = 32 - 11 = 21\]and the series have $5$ elements and we need to find the next i.e., ${6^{th}}$ element, so $n = 6$.
The Arithmetic Progression (AP) formula is
${a_n} = {a_1} + (n - 1)d$
Substituting all the above values in this formula,
${a_6} = 11 + (6 - 1)21$
Solving the bracket first in the above equation,
${a_6} = 11 + (5)21$
Multiplying the terms in the above equation,
${a_6} = 11 + 105$
Adding the terms in the above equation,
${a_6} = 116$
The ${6^{th}}$ element is $116$.
Therefore, the correct option is D. $116$.

Note: After finding the last term, the complete series will be $11,32,53,74,95,116$. This series can continue to be infinite as it can have a number of terms with the same common difference. Series can be of more than two types other than Arithmetic Progression ;they are Geometric and Harmonic Progressions respectively.