Find the sum of first 24 terms of the A.P. a1,a2,a3,… if it is known that a1+a5+a10+a15+a20+a24=225.
Get the complete, step-by-step math solution for: "Find the sum of first 24 terms of the A.P. a_{1}, a_{2}, a_{3}, if it is known that a_{1}+a_{5}+a_{10}+a_{15}+a_{20}+a_{24}=225.". Powered by SolveForX AI math tutor.
Step-by-Step Solution
Step 1: Understand the properties of an A.P.
In an Arithmetic Progression (A.P.), the n -th term is given by the formula an=a+(n−1)d, where a is the first term and d is the common difference. A key property of an A.P. is that the sum of terms equidistant from the beginning and end is constant. For an A.P. with N terms, ak+aN−k+1 is constant.
Step 2: Apply the property of equidistant terms
For an A.P. with 24 terms, the sum of terms equidistant from the beginning and end is equal. Here, a1 and a24 are the first and last terms. a5 is the 5th term and a20 is the 5th term from the end (24−5+1=20). Similarly, a10 is the 10th term and a15 is the 10th term from the end (24−10+1=15). Therefore, a1+a24=a5+a20=a10+a15.
Step 3: Substitute into the given equation
We are given the equation a1+a5+a10+a15+a20+a24=225. Using the property from the previous step, we can group these terms: (a1+a24)+(a5+a20)+(a10+a15)=225. Since each of these pairs sums to the same value, let's call it S′, we have S′+S′+S′=225, or 3S′=225. This gives us S′=a1+a24=75.
Step 4: Calculate the sum of the first 24 terms
The sum of the first n terms of an A.P. is given by the formula Sn=2n(a1+an). For the first 24 terms, n=24. We have already found that a1+a24=75. Substituting these values into the formula, we get S24=224(75)=12×75=900.