If there are $(2 n+1)$ terms in an A.P., then prove that the ratio of the sum of odd terms and the sum of even terms is $(n+1): n$

Step 1: Define the A.P. and its terms

Let the arithmetic progression (A.P.) have $(2n+1)$ terms. We can denote these terms as $a_1, a_2, \dots, a_{2n+1}$. Let $a$ be the first term and $d$ be the common difference of this A.P.

Step 2: Identify odd and even terms

The odd terms are $a_1, a_3, \dots, a_{2n+1}$. There are $(n+1)$ odd terms. The even terms are $a_2, a_4, \dots, a_{2n}$. There are $n$ even terms.

Step 3: Calculate the sum of odd terms

The odd terms form an A.P. with first term $a_1 = a$ and common difference $2d$. There are $(n+1)$ such terms. Using the sum formula for an A.P., $S_k = \frac{k}{2}[2A + (k-1)D]$, where $A$ is the first term and $D$ is the common difference, we find the sum of odd terms.

Step 4: Calculate the sum of even terms

The even terms form an A.P. with first term $a_2 = a+d$ and common difference $2d$. There are $n$ such terms. Applying the sum formula for an A.P., we find the sum of even terms.

Step 5: Find the ratio of the sums

Finally, we find the ratio of the sum of odd terms to the sum of even terms by dividing the expressions we derived. The common factor $(a+nd)$ cancels out, leaving us with the desired ratio.

More from this chapter

• find the number of terms in the AP 7 13 19......205
• is 78 atom of the AP 5 11 17 23
• which time of Ap 3,8,13,18 is 78
• the sum of the first 15 terms of an ap is 450 and its first term is 10 find its common difference
• if the (p)th term of an ap is (q) and the (q)th term is (p) then its (p+q)th term is
• if the teeth term of an ap is Q and the cute term is P then its P + Q to term is
• the fifth time of an ap is 18 and 11th term is 42 find first term
• if the sum of the first n terms of an ap is S sem is equal to 3 and square + 5 and find its common difference

All Arithmetic Progressions solutions →