Arithmetic Progression

Concept: If three terms, say , are in an Arithmetic Progression (A.P.), then the common difference between consecutive terms is constant. This means . This can be rearranged to , which means the middle term is the arithmetic mean of the first and third terms. Step 1: Identify the three terms in A.P. The given terms are:
First term ( ) = K
Second term ( ) = 2K - 1
Third term ( ) = 6 Step 2: Apply the property of A.P. Since the terms are in A.P., the difference between the second and first term is equal to the difference between the third and second term: Alternatively, using the property : Step 3: Solve the equation for K Let's use the second formulation: . Distribute the 2 on the left side: Now, gather K terms on one side and constant terms on the other. Subtract K from both sides: Add 2 to both sides: Divide by 3: Let's verify using the first formulation: Add 2K to both sides: Add 1 to both sides: Both methods yield the same result. The value of K is . This matches option (1).

Concept: In an Arithmetic Progression (AP), the term ( ) is given by the formula , where is the first term and is the common difference. Step 1: Write expressions for the 20th term ( ) and the 15th term ( ) Let be the first term of the AP. Given: Common difference, . For the 20th term ( ): Substitute : For the 15th term ( ): Substitute : Step 2: Calculate the difference The terms cancel out: Step 3: Substitute the value of the common difference ( ) Given : The difference between the 20th and 15th terms is 15. This matches option (3).

Concept: This problem involves finding the sum of terms in two different arithmetic progressions (APs) and then solving an equation based on the ratio of these sums. The sum of an AP is given by , where is the number of terms, is the first term, and is the common difference. Step 1: Find the sum of the numerator AP ( ) The numerator series is to terms. This is an AP with:
First term,
Common difference,
Number of terms = Sum of numerator, Step 2: Find the sum of the denominator AP ( ) The denominator series is to terms. This is an AP with:
First term,
Common difference,
Number of terms = Sum of denominator, Step 3: Set up the equation using the given ratio We are given that . Substitute the expressions for and : Step 4: Solve for Cross-multiply: Expand both sides: Bring all terms to one side to form a quadratic equation: We can solve this quadratic equation or test the given options. Let's test (option 2, which is circled in the image). If : Left side: Since the equation holds true for , this is a valid solution. As represents the number of terms, it must be a positive integer. The other root of this quadratic equation is , which is not a natural number. Thus, is the correct answer. Alternatively, after Step 3, with the equation , we could directly substitute : Numerator for : . Denominator for : . Ratio: . To simplify , we can see if they have common factors. Both are divisible by 7: So, . This matches the given ratio.

Concept: Let be the sum of the first terms of an A.P. Let be the term of the A.P. The term can be found using the relation: (for ). The first term . Alternatively, if is a quadratic in of the form , then it represents the sum of an AP. The common difference and the first term . Then . Method 1: Using Given . We need the 25th term, . So, . Calculate : . Calculate : . Now, calculate : . Method 2: Finding and Given . This is in the form where and . First term, . (Using shortcut: ). Common difference, . The term of an A.P. is . We need the 25th term ( ): . Both methods yield 76.

Concept: The 3-digit natural numbers divisible by 13 form an arithmetic progression (AP). We need to find the first term, the last term, the number of terms, and then use the sum formula for an AP. Sum of an AP: , where is the first term, is the last term, and is the number of terms. Step 1: Find the first 3-digit number divisible by 13 ( ) The smallest 3-digit number is 100. Divide 100 by 13: . So, (2-digit). The next multiple of 13 is . This is the first 3-digit number divisible by 13. So, . Step 2: Find the last 3-digit number divisible by 13 ( ) The largest 3-digit number is 999. Divide 999 by 13: . So, consider . . This is the last 3-digit number divisible by 13. So, . Step 3: Find the number of terms ( ) The terms form an AP: 104, 117, ..., 988 with common difference . Let . . . . . : ; ; bring down 4, so 104. . So, . . There are 69 such numbers. Step 4: Calculate the sum ( ) Using the formula : Calculation of : 546 x 69 ----- 4914 ( ) 32760 ( ) ----- 37674 So, the sum is 37674.

Concept: For an arithmetic progression (AP), the sum of the first terms is given by the formula: where is the first term and is the common difference. First, we need to verify if the given series is indeed an arithmetic progression and find its common difference. Step 1: Expand the terms of the given series Term 1: Term 2: Term 3: Step 2: Calculate the difference between consecutive terms to find the common difference Difference : Difference : Since , the series is an arithmetic progression with: First term, Common difference, Step 3: Calculate the sum of terms ( ) Using the formula : Substitute and : We can factor out a 2 from the terms inside the main bracket: Step 4: Compare the calculated sum with the given options The calculated sum is .
(1) :} This option is identical to our calculated sum .
(2) :} This does not match. It simplifies to .
(3) : This does not resemble the sum formula for this AP.
(4) None of these: Since option (1) matches, this is incorrect. Therefore, option (1) is the correct sum.