Question

Find the missing number in the series:
\[
  {\text{(i) 4,7,13,25,_,97,193 (a)19 (b)21 (c)49 (d)23}} \\
  {\text{(ii) 5,22,90,362,_,5802 (a)1450 (b)1625 (c)3452 (d)2245}} \\
  {\text{(iii) 5,125,25,625,_,3125 (a)1975 (b)1525 (c)875 (d)125}} \\
  {\text{(iv) 5,_,17,29,45,65 (a)7 (b)9 (c)11 (d)13}} \\
 \]

Answer 1

Hint: A series is the sequence of numbers describing the operation of adding, subtracting, multiplying, dividing or any mathematical combination of operation infinitely many quantities one after another arranged in a definite sequence.
To find the missing terms in the series, check the order of the sequence and how the number in the series is dependent upon its previous term or its next term.

Complete step by step answer:
(i) Every number of the sequence is dependent on the next term in a common manner.
\[4,7,13,25,\_,97,193\]
In this series first, we find the difference between each term
\[
  7 - 4 = 3 \\
  13 - 7 = 6 \\
  25 - 13 = 12 \\
 \]
From the difference obtained we can see the difference doubles itself after every number sequence:
\[3 \times 2 \to 6 \times 2 \to 12 \times 2 = 24\]
Hence, the next term in the sequence after \[25\]will be \[25 + 24 = 49\], hence the series is
\[4,7,13,25,\underline {49} ,97,193\],
To check, \[97 - 49 = 48\]which is \[24 \times 2 = 48\]this satisfies the progression.

(ii) In the series, $5,22,90,362,{\text{_,5802}}$ we can see that the difference of each consecutive term is in the series $17,68,272,...$. So, here instead of one series, we have to go with the two series to get the result.
 In the series of $17,68,272,...$, each term is multiplying by a factor of 4 so, the next term after 272 in the series will be :
$17 \times 4 \to 68 \times 4 \to 272 \times 4 \to 1088$
Now, in the first series $5,22,90,362,{\text{_,5802}}$, adding 1088 to 362 will give us the answer:
$5 + 17 \to 22 + 4\left( {17} \right) \to 90 + 4\left( {4\left( {17} \right)} \right) \to 272 + 4\left( {4\left( {4\left( {17} \right)} \right)} \right) \to 1450$
Hence, the series will be $5,22,90,362,1450{\text{,5802}}$.

(iii) Now, for \[5,125,25,625,\_,3125\] we can see each term in the sequence is dependent on its next alternative term,
Where \[5 \times 5 \to 25 \times 5 = 125\]and \[125 \times 5 = 625 \times 5 = 3125\]
We can see each alternative terms are dependent on each other and are in multiple factors of \[5\], hence next term in the series after \[625\] will be \[25 \times 5 = 125\]

Hence, the series is \[5,125,25,625,\underline {125} ,3125\].

(iv) In the series $5,\_\_,17,29,45,65$, as the second term is missing so, we start our calculation from the last term
All the terms from the last term are subtracted by decreasing multiples of 4 i.e., 20,16,12,8,4 so as to get the consecutive term.
$65 - 20 \to 45 - 16 \to 29 - 12 \to 17 - 8 \ to 9$

Hence, the series is $5,9,17,29,45,65$.

Note: To find the missing terms in the series find the nature of the series progression. In an Arithmetic sequence, each number increases or decreases by a common difference. In Geometric Sequence, the series is progressing or decreasing with a multiple ratio. A harmonic sequence is the reciprocal of an Arithmetic sequence. Power sequence is the series where each term is in a common power squared, cubic, etc. In the Alternative series sequence, the progression of the number is dependent on its alternate numbers.