Key Concepts and Formulas

• Arithmetic Progression (AP): A sequence of numbers such that the difference between consecutive terms is constant. The general form is $a, a+d, a+2d,$, where $a$ is the first term and $d$ is the common difference.
• $n^{th}$ Term of an AP: The $n^{th}$ term of an AP is given by $a_n = a + (n-1)d$.
• Common Terms of Two APs: The common terms of two arithmetic progressions themselves form an arithmetic progression.
  • The first term of this new AP is the smallest term that appears in both original APs.
  • The common difference of this new AP is the Least Common Multiple (LCM) of the common differences of the two original APs.

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Step-by-Step Solution

Step 1: Identify the properties of the given arithmetic progressions.

We are given two series, $S_1$ and $S_2$, which are arithmetic progressions. We need to find their first term and common difference.

For the series $S_1 = 3+7+11+15+19+$:

• The first term, $a_1 = 3$.
• The common difference, $d_1 = 7 - 3 = 4$.

For the series $S_2 = 1+6+11+16+21+$:

• The first term, $a_2 = 1$.
• The common difference, $d_2 = 6 - 1 = 5$.

Step 2: Find the first common term of the two series.

The common terms of two APs form a new AP. To find the first term of this new AP, we need to find the smallest number that appears in both $S_1$ and $S_2$. We can do this by listing out the terms of each series.

Terms of $S_1$: $3, 7, **11**, 15, 19, 23, 27, 31,$ Terms of $S_2$: $1, 6, **11**, 16, 21, 26, 31,$

By inspection, the first common term is $11$. Let's denote the first common term as $A = 11$.

Step 3: Determine the common difference of the series of common terms.

The common difference of the AP formed by the common terms is the Least Common Multiple (LCM) of the common differences of the original APs ($d_1$ and $d_2$).

We have $d_1 = 4$ and $d_2 = 5$. The common difference of the common terms, $D$, is LCM$(4, 5)$. Since 4 and 5 are coprime (their greatest common divisor is 1), their LCM is their product: $D = LCM(4, 5) = 4 5 = 20$.

Step 4: Calculate the $8^{th}$ common term.

Now we have the first term ($A = 11$) and the common difference ($D = 20$) of the AP formed by the common terms. We need to find the $8^{th}$ term of this new AP. Using the formula for the $n^{th}$ term of an AP, $a_n = a + (n-1)d$, we have: $A_n = A + (n-1)D$

For the $8^{th}$ common term, $n=8$: $A_8 = 11 + (8-1) 20$ $A_8 = 11 + (7) 20$ $A_8 = 11 + 140$ $A_8 = 151$

Therefore, the $8^{th}$ common term of the two series is $151$.

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Common Mistakes & Tips

• Finding the First Common Term: Do not use a formula to find the first common term. Always list out the initial terms of both series to identify the smallest common value.
• Calculating the Common Difference of Common Terms: Ensure you calculate the LCM of the common differences of the original series. For example, LCM(4, 6) = 12, not $4 6 = 24$.
• Using the Correct 'n': Remember that you are looking for the $8^{th}$ term, so $n$ in the formula $a_n = a + (n-1)d$ should be $8$.

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Summary

To find the $8^{th}$ common term of the two given arithmetic progressions, we first identified the first term and common difference for each series. We then determined the first common term by listing out the initial terms of both series and found it to be $11$. The common difference of the series of common terms was found by taking the LCM of the individual common differences, which resulted in $20$. Finally, using the formula for the $n^{th}$ term of an arithmetic progression, we calculated the $8^{th}$ common term to be $151$.

The final answer is $151$.