The ratio of the 11th term to the 18th term of an A.P. is 2: 3. Find the ratio of the 5th term to the 21th term. Also, find the ratio of the sum of first 5 terms to the sum of first 21 terms

Step 1: Use the given ratio of terms to relate first term and common difference

Let the first term of the arithmetic progression be $a$ and the common difference be $d$. The $n^{\text{th}}$ term is given by $a_n = a + (n-1)d$. Using the given ratio of the 11th term to the 18th term, we cross-multiply to establish an algebraic relation between $a$ and $d$.

Step 2: Express the first term in terms of common difference

Expanding both sides gives $3a + 30d = 2a + 34d$. Subtracting $2a$ and $30d$ from both sides simplifies directly to $a = 4d$.

Step 3: Find the ratio of the 5th term to the 21st term

The 5th term is $a + 4d$ and the 21st term is $a + 20d$. Substituting $a = 4d$ into both expressions, the numerator becomes $8d$ and the denominator becomes $24d$. Dividing gives the ratio $1:3$.

Step 4: Calculate the ratio of the sum of first 5 terms to first 21 terms

The sum of the first $n$ terms is $S_n = \frac{n}{2}[2a + (n-1)d]$. For $n = 5$ and $n = 21$, substitute $a = 4d$ into the numerator and denominator. This yields $\frac{5 \times 12d}{21 \times 28d} = \frac{60}{588}$, which reduces to $\frac{5}{49}$.

More from this chapter

• 'A thief runs away from a police station with a uniform speed of 100 m/minute. After exactly one minute, a policeman runs after the thief to catch him. The poli
• If the sum of the first m terms of an AP is the same as the sum of its first n terms (m ≠ n), show that the sum of its first (m + n) terms is: S_m+n = 0
• Find the 20th term of the AP 7, 11, 15, 19,...
• A school has decided to plant some endangered trees on 51^(st) World Environment Day in the nearest park. They have decided to plant those trees in few concentr
• Two A.P.s have the same first term. The common difference of the first A.P. is -3 and of the second A.P. is -5. The difference of the 6^(th) term of the second
• If k + 7, 2k - 2 and 2k + 6 are three consecutive terms of an A.P., then the value of k is
• (a) Find the sum of all integers between 50 and 500, which are divisible by 7. OR (b) How many numbers lie between 10 and 300, which when divided by 4 leave a r
• If x, 2x + 9, 4x + 3 are three consecutive terms of an A.P., then the value of x is:

All Arithmetic Progressions solutions →